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We define operators $s^{j}$ in terms of $\left\{\mathbf{1}^{R},I^{R},J^{R},K^{R}\right\}$
$$
{}^{1}=\left({\begin{array}{c c}{1^{R}}&{0}\\ {0}&{1^{R}}\end{array}}\right),\quad s^{2}=\left({\begin{array}{c c}{I^{R}}&{0}\\ {0}&{I^{R}}\end{array}}\right),\quad s^{3}=\left({\begin{array}{c c}{J^{R}}&{0}\\ {0}&{J^{R}}\end{array}}\right),\quad s^{4}=\left({\begin{array}{c c}{K^{R}}&{0}\\ {0}&{K^{R}}\end{array}}\right),
$$
In a similar way, the group $S p(2)\sim S p i n(5)$ is the group of quaternion-valued $2\times2$ matrices with unit determinant. We will view $S p(2)$ as acting by left multiplication on a field $\Psi$ in the defining representation. So an element $U\in S p(2)$ acts in the following way:
$$
\Psi\to U\Psi.
$$
Equivalently, in terms of components
$$
\Psi_{a}\to U_{a b}\Psi_{b}.
$$
Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:
$$
\gamma^{1}={\binom{1}{0}}\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{2}={\binom{0}{1}}\,\,\,\,\,\,1{\binom{1}{0}}\,,\,\,\,\,\,\,\,\,\,\,\,\gamma^{3}={\binom{0}{-I}}\,\,\,\,\,\,I\,\,\,\,
$$
$$
\gamma^{4}=\left(\!\begin{array}{c c}{{0}}&{{J}}\\ {{-J}}&{{0}}\end{array}\!\right),\qquad\gamma^{5}=\left(\!\begin{array}{c c}{{0}}&{{K}}\\ {{-K}}&{{0}}\end{array}\!\right).
$$
In turn, $\{I,J,K\}$ can be expressed in terms of the Pauli matrices $\sigma^{i}$
$$
\sigma^{1}=\left(\!\!\begin{array}{c c}{{0}}&{{1}}\\ {{1}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{2}=\left(\!\!\begin{array}{c c}{{0}}&{{-i}}\\ {{i}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{3}=\left(\!\!\begin{array}{c c}{{1}}&{{0}}\\ {{0}}&{{-1}}\end{array}\!\!\right)
$$
as $4\times4$ real anti-symmetric matrices:
$$
\begin{array}{c c c}{{I=\left(\begin{array}{c c}{{0}}&{{\sigma^{1}}}\\ {{-\sigma^{1}}}&{{0}}\end{array}\right),~~}}&{{J=\left(\begin{array}{c c}{{-i\sigma^{2}}}&{{0}}\\ {{0}}&{{-i\sigma^{2}}}\end{array}\right),~~}}&{{K=\left(\begin{array}{c c}{{0}}&{{\sigma^{3}}}\\ {{-\sigma^{3}}}&{{0}}\end{array}\right).}}\end{array}
$$
|
<html><body>
<p data-bbox="69 69 350 86">We define operators $s^{j}$ in terms of $\left\{\mathbf{1}^{R},I^{R},J^{R},K^{R}\right\}$ </p>
<div class="equation" data-bbox="98 101 500 133">$$
{}^{1}=\left({\begin{array}{c c}{1^{R}}&{0}\\ {0}&{1^{R}}\end{array}}\right),\quad s^{2}=\left({\begin{array}{c c}{I^{R}}&{0}\\ {0}&{I^{R}}\end{array}}\right),\quad s^{3}=\left({\begin{array}{c c}{J^{R}}&{0}\\ {0}&{J^{R}}\end{array}}\right),\quad s^{4}=\left({\begin{array}{c c}{K^{R}}&{0}\\ {0}&{K^{R}}\end{array}}\right),
$$</div>
<p data-bbox="70 144 542 198">In a similar way, the group $S p(2)\sim S p i n(5)$ is the group of quaternion-valued $2\times2$ matrices with unit determinant. We will view $S p(2)$ as acting by left multiplication on a field $\Psi$ in the defining representation. So an element $U\in S p(2)$ acts in the following way: </p>
<div class="equation" data-bbox="281 219 330 229">$$
\Psi\to U\Psi.
$$</div>
<p data-bbox="69 249 267 264">Equivalently, in terms of components </p>
<div class="equation" data-bbox="272 286 339 298">$$
\Psi_{a}\to U_{a b}\Psi_{b}.
$$</div>
<p data-bbox="69 315 541 332">Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions: </p>
<div class="equation" data-bbox="153 347 457 377">$$
\gamma^{1}={\binom{1}{0}}\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{2}={\binom{0}{1}}\,\,\,\,\,\,1{\binom{1}{0}}\,,\,\,\,\,\,\,\,\,\,\,\,\gamma^{3}={\binom{0}{-I}}\,\,\,\,\,\,I\,\,\,\,
$$</div>
<div class="equation" data-bbox="197 391 414 421">$$
\gamma^{4}=\left(\!\begin{array}{c c}{{0}}&{{J}}\\ {{-J}}&{{0}}\end{array}\!\right),\qquad\gamma^{5}=\left(\!\begin{array}{c c}{{0}}&{{K}}\\ {{-K}}&{{0}}\end{array}\!\right).
$$</div>
<p data-bbox="70 426 432 442">In turn, $\{I,J,K\}$ can be expressed in terms of the Pauli matrices $\sigma^{i}$ </p>
<div class="equation" data-bbox="154 457 456 488">$$
\sigma^{1}=\left(\!\!\begin{array}{c c}{{0}}&{{1}}\\ {{1}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{2}=\left(\!\!\begin{array}{c c}{{0}}&{{-i}}\\ {{i}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{3}=\left(\!\!\begin{array}{c c}{{1}}&{{0}}\\ {{0}}&{{-1}}\end{array}\!\!\right)
$$</div>
<p data-bbox="69 498 271 514">as $4\times4$ real anti-symmetric matrices: </p>
<div class="equation" data-bbox="123 527 488 558">$$
\begin{array}{c c c}{{I=\left(\begin{array}{c c}{{0}}&{{\sigma^{1}}}\\ {{-\sigma^{1}}}&{{0}}\end{array}\right),~~}}&{{J=\left(\begin{array}{c c}{{-i\sigma^{2}}}&{{0}}\\ {{0}}&{{-i\sigma^{2}}}\end{array}\right),~~}}&{{K=\left(\begin{array}{c c}{{0}}&{{\sigma^{3}}}\\ {{-\sigma^{3}}}&{{0}}\end{array}\right).}}\end{array}
$$</div>
</body></html>
|
0001189v2
| 11 | 612 | 792 | 1,275 | 1,650 |
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We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 498, 271, 514], "lines": [{"bbox": [70, 500, 271, 514], "spans": [{"bbox": [70, 500, 86, 514], "score": 1.0, "content": "as ", "type": "text"}, {"bbox": [86, 503, 113, 512], "score": 0.93, "content": "4\\times4", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [113, 500, 271, 514], "score": 1.0, "content": " real anti-symmetric matrices:", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 743], "lines": [{"bbox": [298, 732, 312, 744], "spans": [{"bbox": [298, 732, 312, 744], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 350, 86], "lines": [{"bbox": [70, 71, 348, 88], "spans": [{"bbox": [70, 71, 179, 88], "score": 1.0, "content": "We define operators ", "type": "text"}, {"bbox": [179, 73, 190, 84], "score": 0.89, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [190, 71, 255, 88], "score": 1.0, "content": " in terms of", "type": "text"}, {"bbox": [255, 73, 348, 88], "score": 0.9, "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "type": "inline_equation", "height": 15, "width": 93}], "index": 0}], "index": 0, "bbox_fs": [70, 71, 348, 88]}, {"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 144, 542, 198], "lines": [{"bbox": [93, 147, 540, 163], "spans": [{"bbox": [93, 147, 241, 163], "score": 1.0, "content": "In a similar way, the group ", "type": "text"}, {"bbox": [242, 147, 329, 160], "score": 0.92, "content": "S p(2)\\sim S p i n(5)", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [330, 147, 512, 163], "score": 1.0, "content": " is the group of quaternion-valued ", "type": "text"}, {"bbox": [513, 149, 540, 158], "score": 0.91, "content": "2\\times2", "type": "inline_equation", "height": 9, "width": 27}], "index": 2}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 316, 182], "score": 1.0, "content": "matrices with unit determinant. We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [70, 147, 541, 201]}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [71, 251, 267, 266]}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [69, 318, 541, 335]}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11, "bbox_fs": [70, 430, 430, 443]}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 498, 271, 514], "lines": [{"bbox": [70, 500, 271, 514], "spans": [{"bbox": [70, 500, 86, 514], "score": 1.0, "content": "as ", "type": "text"}, {"bbox": [86, 503, 113, 512], "score": 0.93, "content": "4\\times4", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [113, 500, 271, 514], "score": 1.0, "content": " real anti-symmetric matrices:", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [70, 500, 271, 514]}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}]}
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[{"bbox": [70, 71, 348, 88], "content": "We define operators in terms of", "parent_index": 0, "line_index": 0}, {"bbox": [93, 147, 540, 163], "content": "In a similar way, the group is the group of quaternion-valued", "parent_index": 2, "line_index": 0}, {"bbox": [70, 165, 541, 182], "content": "matrices with unit determinant. We will view as acting by left multiplication on a", "parent_index": 2, "line_index": 1}, {"bbox": [70, 184, 541, 201], "content": "field in the defining representation. So an element acts in the following way:", "parent_index": 2, "line_index": 2}, {"bbox": [71, 251, 267, 266], "content": "Equivalently, in terms of components", "parent_index": 4, "line_index": 0}, {"bbox": [69, 318, 541, 335], "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "parent_index": 6, "line_index": 0}, {"bbox": [70, 430, 430, 443], "content": "In turn, can be expressed in terms of the Pauli matrices", "parent_index": 9, "line_index": 0}, {"bbox": [70, 500, 271, 514], "content": "as real anti-symmetric matrices:", "parent_index": 11, "line_index": 0}]
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[]
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[{"bbox": [179, 73, 190, 84], "content": "s^{j}", "parent_index": 0, "subtype": "inline"}, {"bbox": [255, 73, 348, 88], "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [98, 101, 500, 133], "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "parent_index": 1, "subtype": "interline"}, {"bbox": [242, 147, 329, 160], "content": "S p(2)\\sim S p i n(5)", "parent_index": 2, "subtype": "inline"}, {"bbox": [513, 149, 540, 158], "content": "2\\times2", "parent_index": 2, "subtype": "inline"}, {"bbox": [317, 167, 346, 180], "content": "S p(2)", "parent_index": 2, "subtype": "inline"}, {"bbox": [97, 187, 106, 195], "content": "\\Psi", "parent_index": 2, "subtype": "inline"}, {"bbox": [349, 186, 403, 199], "content": "U\\in S p(2)", "parent_index": 2, "subtype": "inline"}, {"bbox": [281, 219, 330, 229], "content": "\\Psi\\to U\\Psi.", "parent_index": 3, "subtype": "interline"}, {"bbox": [272, 286, 339, 298], "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "parent_index": 5, "subtype": "interline"}, {"bbox": [153, 347, 457, 377], "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "parent_index": 7, "subtype": "interline"}, {"bbox": [197, 391, 414, 421], "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "parent_index": 8, "subtype": "interline"}, {"bbox": [116, 431, 163, 443], "content": "\\{I,J,K\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [419, 430, 430, 440], "content": "\\sigma^{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [154, 457, 456, 488], "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "parent_index": 10, "subtype": "interline"}, {"bbox": [86, 503, 113, 512], "content": "4\\times4", "parent_index": 11, "subtype": "inline"}, {"bbox": [123, 527, 488, 558], "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "parent_index": 12, "subtype": "interline"}]
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[]
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# References
[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196.
[2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85.
[3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47.
[4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997.
[5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541.
[6] N. Hitchin, math.DG/9909002.
[7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070.
[8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675.
[9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307.
[10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184.
[11] V. G. Kac and A. Smilga, hep-th/9908096.
[12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007.
[13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77.
[14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147.
[15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350.
[16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367.
[17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080.
[18] J. Jost and K. Zuo, “Vanishing Theorems for $L^{2}$ -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).
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<html><body>
<h1 data-bbox="272 70 339 85">References </h1>
<p data-bbox="66 100 543 537">[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196.
[2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85.
[3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47.
[4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997.
[5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541.
[6] N. Hitchin, math.DG/9909002.
[7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070.
[8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675.
[9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307.
[10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184.
[11] V. G. Kac and A. Smilga, hep-th/9908096.
[12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007.
[13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77.
[14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147.
[15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350.
[16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367.
[17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080.
[18] J. Jost and K. Zuo, “Vanishing Theorems for $L^{2}$ -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998). </p>
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# Genus one 1-bridge knots and Dunwoody manifolds∗
Luigi Grasselli Michele Mulazzani
November 1, 2018
# Abstract
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
2000 Mathematics Subject Classification: Primary 57M12, 57M25;
Secondary 20F05, 57M05.
Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams.
# 1 Introduction and preliminaries
The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable conditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented manifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide subclass are proved to be cyclic coverings of $\mathrm{{S^{3}}}$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
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<html><body>
<h1 data-bbox="132 166 477 213">Genus one 1-bridge knots and Dunwoody manifolds∗ </h1>
<p data-bbox="183 231 426 248">Luigi Grasselli Michele Mulazzani </p>
<p data-bbox="249 261 361 276">November 1, 2018 </p>
<h1 data-bbox="280 320 329 333">Abstract </h1>
<p data-bbox="138 340 471 461">In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. </p>
<p data-bbox="139 475 469 502">2000 Mathematics Subject Classification: Primary 57M12, 57M25;
Secondary 20F05, 57M05. </p>
<p data-bbox="138 504 470 543">Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams. </p>
<h1 data-bbox="111 565 399 584">1 Introduction and preliminaries </h1>
<p data-bbox="110 596 500 639">The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable conditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented manifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide subclass are proved to be cyclic coverings of $\mathrm{{S^{3}}}$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. </p>
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0003042v1
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[{"type": "text", "text": "Genus one 1-bridge knots and Dunwoody manifolds∗ ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Luigi Grasselli Michele Mulazzani ", "page_idx": 0}, {"type": "text", "text": "November 1, 2018 ", "page_idx": 0}, {"type": "text", "text": "Abstract ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. ", "page_idx": 0}, {"type": "text", "text": "2000 Mathematics Subject Classification: Primary 57M12, 57M25; \nSecondary 20F05, 57M05. ", "page_idx": 0}, {"type": "text", "text": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams. ", "page_idx": 0}, {"type": "text", "text": "1 Introduction and preliminaries ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable conditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented manifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide subclass are proved to be cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. ", "page_idx": 0}]
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J. Dunwoody are cyclic coverings of lens spaces (eventually ", "type": "text"}, {"bbox": [450, 356, 462, 366], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [462, 356, 470, 369], "score": 1.0, "content": "),", "type": "text"}], "index": 6}, {"bbox": [139, 369, 470, 382], "spans": [{"bbox": [139, 369, 470, 382], "score": 1.0, "content": "branched over genus one 1-bridge knots. As a consequence, we give", "type": "text"}], "index": 7}, {"bbox": [139, 384, 471, 396], "spans": [{"bbox": [139, 384, 471, 396], "score": 1.0, "content": "a positive answer to the Dunwoody conjecture that all the elements", "type": "text"}], "index": 8}, {"bbox": [138, 397, 470, 410], "spans": [{"bbox": [138, 397, 345, 410], "score": 1.0, "content": "of a wide subclass are cyclic coverings of ", "type": "text"}, {"bbox": [345, 397, 358, 406], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [358, 397, 470, 410], "score": 1.0, "content": " branched over a knot.", "type": "text"}], "index": 9}, {"bbox": [138, 410, 470, 423], "spans": [{"bbox": [138, 410, 470, 423], "score": 1.0, "content": "Moreover, we show that all branched cyclic coverings of a 2-bridge", "type": "text"}], "index": 10}, {"bbox": [138, 423, 470, 437], "spans": [{"bbox": [138, 423, 470, 437], "score": 1.0, "content": "knot belong to this subclass; this implies that the fundamental group", "type": "text"}], "index": 11}, {"bbox": [138, 437, 471, 451], "spans": [{"bbox": [138, 437, 471, 451], "score": 1.0, "content": "of each branched cyclic covering of a 2-bridge knot admits a geometric", "type": "text"}], "index": 12}, {"bbox": [139, 451, 232, 464], "spans": [{"bbox": [139, 451, 232, 464], "score": 1.0, "content": "cyclic presentation.", "type": "text"}], "index": 13}], "index": 9, "bbox_fs": [138, 343, 471, 464]}, {"type": "list", "bbox": [139, 475, 469, 502], "lines": [{"bbox": [140, 477, 469, 490], "spans": [{"bbox": [140, 477, 469, 490], "score": 1.0, "content": "2000 Mathematics Subject Classification: Primary 57M12, 57M25;", "type": "text"}], "index": 14, "is_list_end_line": true}, {"bbox": [139, 492, 262, 504], "spans": [{"bbox": [139, 492, 262, 504], "score": 1.0, "content": "Secondary 20F05, 57M05.", "type": "text"}], "index": 15, "is_list_start_line": true, "is_list_end_line": true}], "index": 14.5, "bbox_fs": [139, 477, 469, 504]}, {"type": "text", "bbox": [138, 504, 470, 543], "lines": [{"bbox": [140, 504, 469, 519], "spans": [{"bbox": [140, 504, 469, 519], "score": 1.0, "content": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli-", "type": "text"}], "index": 16}, {"bbox": [138, 518, 470, 532], "spans": [{"bbox": [138, 518, 470, 532], "score": 1.0, "content": "cally presented groups, geometric presentations of groups, Heegaard", "type": "text"}], "index": 17}, {"bbox": [138, 532, 187, 545], "spans": [{"bbox": [138, 532, 187, 545], "score": 1.0, "content": "diagrams.", "type": "text"}], "index": 18}], "index": 17, "bbox_fs": [138, 504, 470, 545]}, {"type": "title", "bbox": [111, 565, 399, 584], "lines": [{"bbox": [110, 568, 401, 585], "spans": [{"bbox": [110, 570, 120, 582], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [136, 568, 401, 585], "score": 1.0, "content": "Introduction and preliminaries", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [110, 596, 500, 639], "lines": [{"bbox": [110, 597, 499, 612], "spans": [{"bbox": [110, 597, 499, 612], "score": 1.0, "content": "The problem of determining if a balanced presentation of a group is geomet-", "type": "text"}], "index": 20}, {"bbox": [110, 612, 500, 626], "spans": [{"bbox": [110, 612, 500, 626], "score": 1.0, "content": "ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is", "type": "text"}], "index": 21}, {"bbox": [110, 626, 500, 641], "spans": [{"bbox": [110, 626, 500, 641], "score": 1.0, "content": "quite important within geometric topology and has been deeply investigated", "type": "text"}], "index": 22}, {"bbox": [109, 127, 501, 143], "spans": [{"bbox": [109, 127, 501, 143], "score": 1.0, "content": "by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 143, 501, 155], "spans": [{"bbox": [110, 143, 501, 155], "score": 1.0, "content": "tions between branched cyclic coverings of links and cyclic presentations of", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [109, 157, 500, 170], "spans": [{"bbox": [109, 157, 500, 170], "score": 1.0, "content": "groups induced by suitable Heegaard diagrams have been recently pointed", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [109, 171, 500, 186], "spans": [{"bbox": [109, 171, 500, 186], "score": 1.0, "content": "out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [109, 185, 500, 200], "spans": [{"bbox": [109, 185, 500, 200], "score": 1.0, "content": "In order to investigate these connections, M.J. Dunwoody introduces in [6]", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 200, 500, 214], "spans": [{"bbox": [109, 200, 500, 214], "score": 1.0, "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 215, 500, 228], "spans": [{"bbox": [110, 215, 500, 228], "score": 1.0, "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 500, 257], "score": 1.0, "content": "such that the presentation of the fundamental group of the represented man-", "type": "text", "cross_page": true}], "index": 8}, {"bbox": [109, 258, 500, 271], "spans": [{"bbox": [109, 258, 500, 271], "score": 1.0, "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [109, 272, 501, 286], "spans": [{"bbox": [109, 272, 501, 286], "score": 1.0, "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "type": "text", "cross_page": true}], "index": 10}, {"bbox": [110, 288, 500, 301], "spans": [{"bbox": [110, 288, 500, 301], "score": 1.0, "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [109, 301, 501, 316], "spans": [{"bbox": [109, 301, 501, 316], "score": 1.0, "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [109, 315, 500, 330], "spans": [{"bbox": [109, 315, 500, 330], "score": 1.0, "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "type": "text", "cross_page": true}], "index": 13}, {"bbox": [109, 329, 500, 344], "spans": [{"bbox": [109, 329, 500, 344], "score": 1.0, "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "type": "text", "cross_page": true}], "index": 14}, {"bbox": [109, 344, 500, 359], "spans": [{"bbox": [109, 344, 326, 359], "score": 1.0, "content": "class are proved to be cyclic coverings of ", "type": "text", "cross_page": true}, {"bbox": [326, 345, 339, 355], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13, "cross_page": true}, {"bbox": [339, 344, 500, 359], "score": 1.0, "content": ", branched over suitable knots,", "type": "text", "cross_page": true}], "index": 15}, {"bbox": [110, 360, 500, 373], "spans": [{"bbox": [110, 360, 500, 373], "score": 1.0, "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "type": "text", "cross_page": true}], "index": 16}, {"bbox": [110, 373, 500, 388], "spans": [{"bbox": [110, 373, 500, 388], "score": 1.0, "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "type": "text", "cross_page": true}], "index": 17}, {"bbox": [110, 388, 499, 401], "spans": [{"bbox": [110, 388, 499, 401], "score": 1.0, "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "type": "text", "cross_page": true}], "index": 18}, {"bbox": [110, 402, 500, 416], "spans": [{"bbox": [110, 402, 500, 416], "score": 1.0, "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "type": "text", "cross_page": true}], "index": 19}, {"bbox": [110, 417, 262, 431], "spans": [{"bbox": [110, 417, 262, 431], "score": 1.0, "content": "geometric cyclic presentation.", "type": "text", "cross_page": true}], "index": 20}], "index": 21, "bbox_fs": [110, 597, 500, 641]}]}
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[{"type": "title", "bbox": [132, 166, 477, 213], "content": "Genus one 1-bridge knots and Dunwoody manifolds∗", "index": 0}, {"type": "text", "bbox": [183, 231, 426, 248], "content": "Luigi Grasselli Michele Mulazzani", "index": 1}, {"type": "text", "bbox": [249, 261, 361, 276], "content": "November 1, 2018", "index": 2}, {"type": "title", "bbox": [280, 320, 329, 333], "content": "Abstract", "index": 3}, {"type": "text", "bbox": [138, 340, 471, 461], "content": "In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.", "index": 4}, {"type": "list", "bbox": [139, 475, 469, 502], "content": "", "index": 5}, {"type": "text", "bbox": [138, 504, 470, 543], "content": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.", "index": 6}, {"type": "title", "bbox": [111, 565, 399, 584], "content": "1 Introduction and preliminaries", "index": 7}, {"type": "text", "bbox": [110, 596, 500, 639], "content": "The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.", "index": 8}]
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[{"bbox": [133, 169, 476, 190], "content": "Genus one 1-bridge knots and Dunwoody", "parent_index": 0, "line_index": 0}, {"bbox": [262, 195, 352, 213], "content": "manifolds∗", "parent_index": 0, "line_index": 1}, {"bbox": [183, 234, 426, 249], "content": "Luigi Grasselli Michele Mulazzani", "parent_index": 1, "line_index": 0}, {"bbox": [249, 263, 361, 277], "content": "November 1, 2018", "parent_index": 2, "line_index": 0}, {"bbox": [280, 321, 329, 334], "content": "Abstract", "parent_index": 3, "line_index": 0}, {"bbox": [155, 343, 470, 354], "content": "In this paper we show that all 3-manifolds of a family introduced", "parent_index": 4, "line_index": 0}, {"bbox": [139, 356, 470, 369], "content": "by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ),", "parent_index": 4, "line_index": 1}, {"bbox": [139, 369, 470, 382], "content": "branched over genus one 1-bridge knots. As a consequence, we give", "parent_index": 4, "line_index": 2}, {"bbox": [139, 384, 471, 396], "content": "a positive answer to the Dunwoody conjecture that all the elements", "parent_index": 4, "line_index": 3}, {"bbox": [138, 397, 470, 410], "content": "of a wide subclass are cyclic coverings of branched over a knot.", "parent_index": 4, "line_index": 4}, {"bbox": [138, 410, 470, 423], "content": "Moreover, we show that all branched cyclic coverings of a 2-bridge", "parent_index": 4, "line_index": 5}, {"bbox": [138, 423, 470, 437], "content": "knot belong to this subclass; this implies that the fundamental group", "parent_index": 4, "line_index": 6}, {"bbox": [138, 437, 471, 451], "content": "of each branched cyclic covering of a 2-bridge knot admits a geometric", "parent_index": 4, "line_index": 7}, {"bbox": [139, 451, 232, 464], "content": "cyclic presentation.", "parent_index": 4, "line_index": 8}, {"bbox": [140, 477, 469, 490], "content": "2000 Mathematics Subject Classification: Primary 57M12, 57M25;", "parent_index": 5, "line_index": 0}, {"bbox": [139, 492, 262, 504], "content": "Secondary 20F05, 57M05.", "parent_index": 5, "line_index": 1}, {"bbox": [140, 504, 469, 519], "content": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli-", "parent_index": 6, "line_index": 0}, {"bbox": [138, 518, 470, 532], "content": "cally presented groups, geometric presentations of groups, Heegaard", "parent_index": 6, "line_index": 1}, {"bbox": [138, 532, 187, 545], "content": "diagrams.", "parent_index": 6, "line_index": 2}, {"bbox": [110, 568, 401, 585], "content": "1 Introduction and preliminaries", "parent_index": 7, "line_index": 0}, {"bbox": [110, 597, 499, 612], "content": "The problem of determining if a balanced presentation of a group is geomet-", "parent_index": 8, "line_index": 0}, {"bbox": [110, 612, 500, 626], "content": "ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is", "parent_index": 8, "line_index": 1}, {"bbox": [110, 626, 500, 641], "content": "quite important within geometric topology and has been deeply investigated", "parent_index": 8, "line_index": 2}, {"bbox": [109, 127, 501, 143], "content": "by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec-", "parent_index": 8, "line_index": 3}, {"bbox": [110, 143, 501, 155], "content": "tions between branched cyclic coverings of links and cyclic presentations of", "parent_index": 8, "line_index": 4}, {"bbox": [109, 157, 500, 170], "content": "groups induced by suitable Heegaard diagrams have been recently pointed", "parent_index": 8, "line_index": 5}, {"bbox": [109, 171, 500, 186], "content": "out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).", "parent_index": 8, "line_index": 6}, {"bbox": [109, 185, 500, 200], "content": "In order to investigate these connections, M.J. Dunwoody introduces in [6]", "parent_index": 8, "line_index": 7}, {"bbox": [109, 200, 500, 214], "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "parent_index": 8, "line_index": 8}, {"bbox": [110, 215, 500, 228], "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "parent_index": 8, "line_index": 9}, {"bbox": [110, 230, 500, 243], "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "parent_index": 8, "line_index": 10}, {"bbox": [110, 244, 500, 257], "content": "such that the presentation of the fundamental group of the represented man-", "parent_index": 8, "line_index": 11}, {"bbox": [109, 258, 500, 271], "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "parent_index": 8, "line_index": 12}, {"bbox": [109, 272, 501, 286], "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "parent_index": 8, "line_index": 13}, {"bbox": [110, 288, 500, 301], "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "parent_index": 8, "line_index": 14}, {"bbox": [109, 301, 501, 316], "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "parent_index": 8, "line_index": 15}, {"bbox": [109, 315, 500, 330], "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "parent_index": 8, "line_index": 16}, {"bbox": [109, 329, 500, 344], "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "parent_index": 8, "line_index": 17}, {"bbox": [109, 344, 500, 359], "content": "class are proved to be cyclic coverings of , branched over suitable knots,", "parent_index": 8, "line_index": 18}, {"bbox": [110, 360, 500, 373], "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "parent_index": 8, "line_index": 19}, {"bbox": [110, 373, 500, 388], "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "parent_index": 8, "line_index": 20}, {"bbox": [110, 388, 499, 401], "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "parent_index": 8, "line_index": 21}, {"bbox": [110, 402, 500, 416], "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "parent_index": 8, "line_index": 22}, {"bbox": [110, 417, 262, 431], "content": "geometric cyclic presentation.", "parent_index": 8, "line_index": 23}]
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[{"bbox": [450, 356, 462, 366], "content": "\\mathbf{S^{3}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [345, 397, 358, 406], "content": "\\mathbf{S^{3}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [326, 345, 339, 355], "content": "\\mathrm{{S^{3}}}", "parent_index": 8, "subtype": "inline"}]
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For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].
We recall that a finite balanced presentation of a group $<$ $x_{1},\dots\,,x_{n}|r_{1},...\ ,r_{n}\,>$ is said to be a cyclic presentation if there exists a word $w$ in the free group $F_{n}$ generated by $x_{1},\ldots,x_{n}$ such that the relators of the presentation are $r_{k}=\theta_{n}^{k-1}(w)$ , $k=1,\dotsc,n$ , where $\theta_{n}:F_{n}\to F_{n}$ denotes the automorphism defined by $\theta_{n}(x_{i})\,=\,x_{i+1}$ (mod $n$ ), $i=1,\dots,n$ . Let us denote this cyclic presentation (and the related group) by the symbol $G_{n}(w)$ , so that:
$$
G_{n}(w)=<x_{1},x_{2},\ldots,x_{n}|w,\theta_{n}(w),\ldots,\theta_{n}^{n-1}(w)>.
$$
A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $w\,\in\,F_{n}$ is the integer $\varepsilon_{w}$ given by the sum of the exponents of its letters; in other terms, $\varepsilon_{w}=\upsilon(w)$ where $\upsilon:F_{n}\to\mathbf{Z}$ is the homomorphism defined by $\upsilon(x_{i})=1$ for each $1\leq i\leq n$ .
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<html><body>
<p data-bbox="109 429 501 487">For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15]. </p>
<p data-bbox="109 487 500 587">We recall that a finite balanced presentation of a group $<$ $x_{1},\dots\,,x_{n}|r_{1},...\ ,r_{n}\,>$ is said to be a cyclic presentation if there exists a word $w$ in the free group $F_{n}$ generated by $x_{1},\ldots,x_{n}$ such that the relators of the presentation are $r_{k}=\theta_{n}^{k-1}(w)$ , $k=1,\dotsc,n$ , where $\theta_{n}:F_{n}\to F_{n}$ denotes the automorphism defined by $\theta_{n}(x_{i})\,=\,x_{i+1}$ (mod $n$ ), $i=1,\dots,n$ . Let us denote this cyclic presentation (and the related group) by the symbol $G_{n}(w)$ , so that: </p>
<div class="equation" data-bbox="174 589 435 604">$$
G_{n}(w)=<x_{1},x_{2},\ldots,x_{n}|w,\theta_{n}(w),\ldots,\theta_{n}^{n-1}(w)>.
$$</div>
<p data-bbox="110 609 500 667">A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $w\,\in\,F_{n}$ is the integer $\varepsilon_{w}$ given by the sum of the exponents of its letters; in other terms, $\varepsilon_{w}=\upsilon(w)$ where $\upsilon:F_{n}\to\mathbf{Z}$ is the homomorphism defined by $\upsilon(x_{i})=1$ for each $1\leq i\leq n$ . </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15]. ", "page_idx": 1}, {"type": "text", "text": "We recall that a finite balanced presentation of a group $<$ $x_{1},\\dots\\,,x_{n}|r_{1},...\\ ,r_{n}\\,>$ is said to be a cyclic presentation if there exists a word $w$ in the free group $F_{n}$ generated by $x_{1},\\ldots,x_{n}$ such that the relators of the presentation are $r_{k}=\\theta_{n}^{k-1}(w)$ , $k=1,\\dotsc,n$ , where $\\theta_{n}:F_{n}\\to F_{n}$ denotes the automorphism defined by $\\theta_{n}(x_{i})\\,=\\,x_{i+1}$ (mod $n$ ), $i=1,\\dots,n$ . Let us denote this cyclic presentation (and the related group) by the symbol $G_{n}(w)$ , so that: ", "page_idx": 1}, {"type": "equation", "text": "$$\nG_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $w\\,\\in\\,F_{n}$ is the integer $\\varepsilon_{w}$ given by the sum of the exponents of its letters; in other terms, $\\varepsilon_{w}=\\upsilon(w)$ where $\\upsilon:F_{n}\\to\\mathbf{Z}$ is the homomorphism defined by $\\upsilon(x_{i})=1$ for each $1\\leq i\\leq n$ . ", "page_idx": 1}]
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{"preproc_blocks": [{"type": "text", "bbox": [109, 124, 501, 428], "lines": [{"bbox": [109, 127, 501, 143], "spans": [{"bbox": [109, 127, 501, 143], "score": 1.0, "content": "by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec-", "type": "text"}], "index": 0}, {"bbox": [110, 143, 501, 155], "spans": [{"bbox": [110, 143, 501, 155], "score": 1.0, "content": "tions between branched cyclic coverings of links and cyclic presentations of", "type": "text"}], "index": 1}, {"bbox": [109, 157, 500, 170], "spans": [{"bbox": [109, 157, 500, 170], "score": 1.0, "content": "groups induced by suitable Heegaard diagrams have been recently pointed", "type": "text"}], "index": 2}, {"bbox": [109, 171, 500, 186], "spans": [{"bbox": [109, 171, 500, 186], "score": 1.0, "content": "out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 200], "spans": [{"bbox": [109, 185, 500, 200], "score": 1.0, "content": "In order to investigate these connections, M.J. Dunwoody introduces in [6]", "type": "text"}], "index": 4}, {"bbox": [109, 200, 500, 214], "spans": [{"bbox": [109, 200, 500, 214], "score": 1.0, "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "type": "text"}], "index": 5}, {"bbox": [110, 215, 500, 228], "spans": [{"bbox": [110, 215, 500, 228], "score": 1.0, "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "type": "text"}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 500, 257], "score": 1.0, "content": "such that the presentation of the fundamental group of the represented man-", "type": "text"}], "index": 8}, {"bbox": [109, 258, 500, 271], "spans": [{"bbox": [109, 258, 500, 271], "score": 1.0, "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "type": "text"}], "index": 9}, {"bbox": [109, 272, 501, 286], "spans": [{"bbox": [109, 272, 501, 286], "score": 1.0, "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "type": "text"}], "index": 10}, {"bbox": [110, 288, 500, 301], "spans": [{"bbox": [110, 288, 500, 301], "score": 1.0, "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "type": "text"}], "index": 11}, {"bbox": [109, 301, 501, 316], "spans": [{"bbox": [109, 301, 501, 316], "score": 1.0, "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "type": "text"}], "index": 12}, {"bbox": [109, 315, 500, 330], "spans": [{"bbox": [109, 315, 500, 330], "score": 1.0, "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "type": "text"}], "index": 13}, {"bbox": [109, 329, 500, 344], "spans": [{"bbox": [109, 329, 500, 344], "score": 1.0, "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "type": "text"}], "index": 14}, {"bbox": [109, 344, 500, 359], "spans": [{"bbox": [109, 344, 326, 359], "score": 1.0, "content": "class are proved to be cyclic coverings of ", "type": "text"}, {"bbox": [326, 345, 339, 355], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [339, 344, 500, 359], "score": 1.0, "content": ", branched over suitable knots,", "type": "text"}], "index": 15}, {"bbox": [110, 360, 500, 373], "spans": [{"bbox": [110, 360, 500, 373], "score": 1.0, "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "type": "text"}], "index": 16}, {"bbox": [110, 373, 500, 388], "spans": [{"bbox": [110, 373, 500, 388], "score": 1.0, "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "type": "text"}], "index": 17}, {"bbox": [110, 388, 499, 401], "spans": [{"bbox": [110, 388, 499, 401], "score": 1.0, "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "type": "text"}], "index": 18}, {"bbox": [110, 402, 500, 416], "spans": [{"bbox": [110, 402, 500, 416], "score": 1.0, "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "type": "text"}], "index": 19}, {"bbox": [110, 417, 262, 431], "spans": [{"bbox": [110, 417, 262, 431], "score": 1.0, "content": "geometric cyclic presentation.", "type": "text"}], "index": 20}], "index": 10}, {"type": "text", "bbox": [109, 429, 501, 487], "lines": [{"bbox": [127, 431, 498, 444], "spans": [{"bbox": [127, 431, 498, 444], "score": 1.0, "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular", "type": "text"}], "index": 21}, {"bbox": [109, 445, 500, 460], "spans": [{"bbox": [109, 445, 500, 460], "score": 1.0, "content": "for Singer moves on Heegaard diagrams realizing the homeomorphism of the", "type": "text"}], "index": 22}, {"bbox": [110, 461, 498, 474], "spans": [{"bbox": [110, 461, 498, 474], "score": 1.0, "content": "represented manifolds, we refer to [13] and [31]. For the theory of cyclically", "type": "text"}], "index": 23}, {"bbox": [109, 474, 284, 488], "spans": [{"bbox": [109, 474, 284, 488], "score": 1.0, "content": "presented groups, we refer to [15].", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [109, 487, 500, 587], "lines": [{"bbox": [126, 487, 499, 503], "spans": [{"bbox": [126, 487, 489, 503], "score": 1.0, "content": "We recall that a finite balanced presentation of a group", "type": "text"}, {"bbox": [489, 492, 499, 500], "score": 0.36, "content": "<", "type": "inline_equation", "height": 8, "width": 10}], "index": 25}, {"bbox": [110, 504, 502, 516], "spans": [{"bbox": [110, 504, 229, 516], "score": 0.9, "content": "x_{1},\\dots\\,,x_{n}|r_{1},...\\ ,r_{n}\\,>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [229, 505, 502, 516], "score": 1.0, "content": " is said to be a cyclic presentation if there exists a", "type": "text"}], "index": 26}, {"bbox": [109, 517, 503, 533], "spans": [{"bbox": [109, 517, 138, 533], "score": 1.0, "content": "word ", "type": "text"}, {"bbox": [138, 522, 147, 528], "score": 0.88, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [147, 517, 237, 533], "score": 1.0, "content": " in the free group ", "type": "text"}, {"bbox": [237, 519, 250, 530], "score": 0.92, "content": "F_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [251, 517, 322, 533], "score": 1.0, "content": " generated by ", "type": "text"}, {"bbox": [322, 522, 374, 531], "score": 0.9, "content": "x_{1},\\ldots,x_{n}", "type": "inline_equation", "height": 9, "width": 52}, {"bbox": [375, 517, 503, 533], "score": 1.0, "content": " such that the relators of", "type": "text"}], "index": 27}, {"bbox": [108, 530, 501, 547], "spans": [{"bbox": [108, 530, 215, 547], "score": 1.0, "content": "the presentation are ", "type": "text"}, {"bbox": [215, 532, 281, 545], "score": 0.94, "content": "r_{k}=\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [281, 530, 287, 547], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [287, 534, 351, 545], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [352, 530, 390, 547], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [390, 534, 457, 544], "score": 0.94, "content": "\\theta_{n}:F_{n}\\to F_{n}", "type": "inline_equation", "height": 10, "width": 67}, {"bbox": [457, 530, 501, 547], "score": 1.0, "content": " denotes", "type": "text"}], "index": 28}, {"bbox": [110, 547, 501, 561], "spans": [{"bbox": [110, 547, 267, 561], "score": 1.0, "content": "the automorphism defined by ", "type": "text"}, {"bbox": [267, 547, 336, 560], "score": 0.92, "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [336, 547, 372, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [372, 551, 380, 557], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [380, 547, 392, 561], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [392, 549, 456, 559], "score": 0.91, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [456, 547, 501, 561], "score": 1.0, "content": ". Let us", "type": "text"}], "index": 29}, {"bbox": [109, 560, 499, 575], "spans": [{"bbox": [109, 560, 462, 575], "score": 1.0, "content": "denote this cyclic presentation (and the related group) by the symbol ", "type": "text"}, {"bbox": [463, 562, 496, 574], "score": 0.94, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [496, 560, 499, 575], "score": 1.0, "content": ",", "type": "text"}], "index": 30}, {"bbox": [109, 576, 151, 588], "spans": [{"bbox": [109, 576, 151, 588], "score": 1.0, "content": "so that:", "type": "text"}], "index": 31}], "index": 28}, {"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 609, 500, 667], "lines": [{"bbox": [127, 612, 498, 624], "spans": [{"bbox": [127, 612, 498, 624], "score": 1.0, "content": "A group is said to be cyclically presented if it admits a cyclic presentation.", "type": "text"}], "index": 33}, {"bbox": [110, 624, 500, 640], "spans": [{"bbox": [110, 624, 338, 640], "score": 1.0, "content": "We recall that the exponent-sum of a word ", "type": "text"}, {"bbox": [339, 627, 378, 637], "score": 0.94, "content": "w\\,\\in\\,F_{n}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [378, 624, 455, 640], "score": 1.0, "content": " is the integer ", "type": "text"}, {"bbox": [455, 630, 468, 637], "score": 0.9, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [468, 624, 500, 640], "score": 1.0, "content": " given", "type": "text"}], "index": 34}, {"bbox": [110, 640, 500, 654], "spans": [{"bbox": [110, 640, 411, 654], "score": 1.0, "content": "by the sum of the exponents of its letters; in other terms, ", "type": "text"}, {"bbox": [411, 641, 465, 653], "score": 0.95, "content": "\\varepsilon_{w}=\\upsilon(w)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [465, 640, 500, 654], "score": 1.0, "content": " where", "type": "text"}], "index": 35}, {"bbox": [110, 654, 489, 668], "spans": [{"bbox": [110, 656, 167, 666], "score": 0.93, "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [167, 654, 342, 668], "score": 1.0, "content": " is the homomorphism defined by ", "type": "text"}, {"bbox": [342, 655, 389, 668], "score": 0.95, "content": "\\upsilon(x_{i})=1", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [390, 654, 437, 668], "score": 1.0, "content": " for each ", "type": "text"}, {"bbox": [438, 656, 486, 666], "score": 0.92, "content": "1\\leq i\\leq n", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [487, 654, 489, 668], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 124, 501, 428], "lines": [], "index": 10, "bbox_fs": [109, 127, 501, 431], "lines_deleted": true}, {"type": "text", "bbox": [109, 429, 501, 487], "lines": [{"bbox": [127, 431, 498, 444], "spans": [{"bbox": [127, 431, 498, 444], "score": 1.0, "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular", "type": "text"}], "index": 21}, {"bbox": [109, 445, 500, 460], "spans": [{"bbox": [109, 445, 500, 460], "score": 1.0, "content": "for Singer moves on Heegaard diagrams realizing the homeomorphism of the", "type": "text"}], "index": 22}, {"bbox": [110, 461, 498, 474], "spans": [{"bbox": [110, 461, 498, 474], "score": 1.0, "content": "represented manifolds, we refer to [13] and [31]. For the theory of cyclically", "type": "text"}], "index": 23}, {"bbox": [109, 474, 284, 488], "spans": [{"bbox": [109, 474, 284, 488], "score": 1.0, "content": "presented groups, we refer to [15].", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [109, 431, 500, 488]}, {"type": "text", "bbox": [109, 487, 500, 587], "lines": [{"bbox": [126, 487, 499, 503], "spans": [{"bbox": [126, 487, 489, 503], "score": 1.0, "content": "We recall that a finite balanced presentation of a group", "type": "text"}, {"bbox": [489, 492, 499, 500], "score": 0.36, "content": "<", "type": "inline_equation", "height": 8, "width": 10}], "index": 25}, {"bbox": [110, 504, 502, 516], "spans": [{"bbox": [110, 504, 229, 516], "score": 0.9, "content": "x_{1},\\dots\\,,x_{n}|r_{1},...\\ ,r_{n}\\,>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [229, 505, 502, 516], "score": 1.0, "content": " is said to be a cyclic presentation if there exists a", "type": "text"}], "index": 26}, {"bbox": [109, 517, 503, 533], "spans": [{"bbox": [109, 517, 138, 533], "score": 1.0, "content": "word ", "type": "text"}, {"bbox": [138, 522, 147, 528], "score": 0.88, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [147, 517, 237, 533], "score": 1.0, "content": " in the free group ", "type": "text"}, {"bbox": [237, 519, 250, 530], "score": 0.92, "content": "F_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [251, 517, 322, 533], "score": 1.0, "content": " generated by ", "type": "text"}, {"bbox": [322, 522, 374, 531], "score": 0.9, "content": "x_{1},\\ldots,x_{n}", "type": "inline_equation", "height": 9, "width": 52}, {"bbox": [375, 517, 503, 533], "score": 1.0, "content": " such that the relators of", "type": "text"}], "index": 27}, {"bbox": [108, 530, 501, 547], "spans": [{"bbox": [108, 530, 215, 547], "score": 1.0, "content": "the presentation are ", "type": "text"}, {"bbox": [215, 532, 281, 545], "score": 0.94, "content": "r_{k}=\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [281, 530, 287, 547], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [287, 534, 351, 545], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [352, 530, 390, 547], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [390, 534, 457, 544], "score": 0.94, "content": "\\theta_{n}:F_{n}\\to F_{n}", "type": "inline_equation", "height": 10, "width": 67}, {"bbox": [457, 530, 501, 547], "score": 1.0, "content": " denotes", "type": "text"}], "index": 28}, {"bbox": [110, 547, 501, 561], "spans": [{"bbox": [110, 547, 267, 561], "score": 1.0, "content": "the automorphism defined by ", "type": "text"}, {"bbox": [267, 547, 336, 560], "score": 0.92, "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [336, 547, 372, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [372, 551, 380, 557], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [380, 547, 392, 561], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [392, 549, 456, 559], "score": 0.91, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [456, 547, 501, 561], "score": 1.0, "content": ". Let us", "type": "text"}], "index": 29}, {"bbox": [109, 560, 499, 575], "spans": [{"bbox": [109, 560, 462, 575], "score": 1.0, "content": "denote this cyclic presentation (and the related group) by the symbol ", "type": "text"}, {"bbox": [463, 562, 496, 574], "score": 0.94, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [496, 560, 499, 575], "score": 1.0, "content": ",", "type": "text"}], "index": 30}, {"bbox": [109, 576, 151, 588], "spans": [{"bbox": [109, 576, 151, 588], "score": 1.0, "content": "so that:", "type": "text"}], "index": 31}], "index": 28, "bbox_fs": [108, 487, 503, 588]}, {"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 609, 500, 667], "lines": [{"bbox": [127, 612, 498, 624], "spans": [{"bbox": [127, 612, 498, 624], "score": 1.0, "content": "A group is said to be cyclically presented if it admits a cyclic presentation.", "type": "text"}], "index": 33}, {"bbox": [110, 624, 500, 640], "spans": [{"bbox": [110, 624, 338, 640], "score": 1.0, "content": "We recall that the exponent-sum of a word ", "type": "text"}, {"bbox": [339, 627, 378, 637], "score": 0.94, "content": "w\\,\\in\\,F_{n}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [378, 624, 455, 640], "score": 1.0, "content": " is the integer ", "type": "text"}, {"bbox": [455, 630, 468, 637], "score": 0.9, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [468, 624, 500, 640], "score": 1.0, "content": " given", "type": "text"}], "index": 34}, {"bbox": [110, 640, 500, 654], "spans": [{"bbox": [110, 640, 411, 654], "score": 1.0, "content": "by the sum of the exponents of its letters; in other terms, ", "type": "text"}, {"bbox": [411, 641, 465, 653], "score": 0.95, "content": "\\varepsilon_{w}=\\upsilon(w)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [465, 640, 500, 654], "score": 1.0, "content": " where", "type": "text"}], "index": 35}, {"bbox": [110, 654, 489, 668], "spans": [{"bbox": [110, 656, 167, 666], "score": 0.93, "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [167, 654, 342, 668], "score": 1.0, "content": " is the homomorphism defined by ", "type": "text"}, {"bbox": [342, 655, 389, 668], "score": 0.95, "content": "\\upsilon(x_{i})=1", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [390, 654, 437, 668], "score": 1.0, "content": " for each ", "type": "text"}, {"bbox": [438, 656, 486, 666], "score": 0.92, "content": "1\\leq i\\leq n", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [487, 654, 489, 668], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34.5, "bbox_fs": [110, 612, 500, 668]}]}
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[{"type": "text", "bbox": [109, 124, 501, 428], "content": "", "index": 0}, {"type": "text", "bbox": [109, 429, 501, 487], "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].", "index": 1}, {"type": "text", "bbox": [109, 487, 500, 587], "content": "We recall that a finite balanced presentation of a group is said to be a cyclic presentation if there exists a word in the free group generated by such that the relators of the presentation are , , where denotes the automorphism defined by (mod ), . Let us denote this cyclic presentation (and the related group) by the symbol , so that:", "index": 2}, {"type": "interline_equation", "bbox": [174, 589, 435, 604], "content": "", "index": 3}, {"type": "text", "bbox": [110, 609, 500, 667], "content": "A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word is the integer given by the sum of the exponents of its letters; in other terms, where is the homomorphism defined by for each .", "index": 4}]
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Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\mathrm{{S^{3}}}$ (see [5]).
A set of mutually disjoint arcs $\{t_{1},\ldots,t_{n}\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\,=$ $\{D_{1},...\,,D_{n}\}$ such that $t_{i}\cap D_{i}=t_{i}\cap\partial D_{i}=t_{i}$ , $t_{i}\cap D_{j}=\emptyset$ and $\partial D_{i}-t_{i}\subset\partial U$ for $1\leq i,j\leq n$ and $i\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot.
The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\prime}$ is defined only for $g\geq g^{\prime}$ and the genus 0 bridge number of a link in $\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\geq0$ .
In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23].
# 2 Dunwoody manifolds
Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\ >\ 0$ , $a,b,c\,\geq\,0$ and $a+b+c>0$ . Let $\Gamma=\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1.
It contains $n$ upper cycles $C_{1}^{\prime},\ldots\,,C_{n}^{\prime}$ and $n$ lower cycles $C_{1}^{\prime\prime},\ldots\,,C_{n}^{\prime\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\dots,n$ , the cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) is connected to the cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\prime\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\prime\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\mathcal{C}^{\prime}=\{C_{1}^{\prime},\ldots,C_{n}^{\prime}\}$ and $\mathcal{C}^{\prime\prime}=\{C_{1}^{\prime\prime},\ldots,C_{n}^{\prime\prime}\}$ . Moreover, denote by $A^{\prime}$ (resp. $A^{\prime\prime}$ ) the set of the arcs of $\Gamma$ belonging to a cycle of $\mathcal{C}^{\prime}$ (resp. $\mathcal{C^{\prime\prime}}$ ) and by
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<p data-bbox="109 125 500 168">Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\mathrm{{S^{3}}}$ (see [5]). </p>
<p data-bbox="109 169 500 298">A set of mutually disjoint arcs $\{t_{1},\ldots,t_{n}\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\,=$ $\{D_{1},...\,,D_{n}\}$ such that $t_{i}\cap D_{i}=t_{i}\cap\partial D_{i}=t_{i}$ , $t_{i}\cap D_{j}=\emptyset$ and $\partial D_{i}-t_{i}\subset\partial U$ for $1\leq i,j\leq n$ and $i\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot. </p>
<p data-bbox="109 299 500 399">The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\prime}$ is defined only for $g\geq g^{\prime}$ and the genus 0 bridge number of a link in $\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\geq0$ . </p>
<p data-bbox="109 400 500 473">In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23]. </p>
<h1 data-bbox="110 493 318 513">2 Dunwoody manifolds </h1>
<p data-bbox="109 524 500 567">Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\ >\ 0$ , $a,b,c\,\geq\,0$ and $a+b+c>0$ . Let $\Gamma=\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1. </p>
<p data-bbox="110 579 500 667">It contains $n$ upper cycles $C_{1}^{\prime},\ldots\,,C_{n}^{\prime}$ and $n$ lower cycles $C_{1}^{\prime\prime},\ldots\,,C_{n}^{\prime\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\dots,n$ , the cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) is connected to the cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\prime\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\prime\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\mathcal{C}^{\prime}=\{C_{1}^{\prime},\ldots,C_{n}^{\prime}\}$ and $\mathcal{C}^{\prime\prime}=\{C_{1}^{\prime\prime},\ldots,C_{n}^{\prime\prime}\}$ . Moreover, denote by $A^{\prime}$ (resp. $A^{\prime\prime}$ ) the set of the arcs of $\Gamma$ belonging to a cycle of $\mathcal{C}^{\prime}$ (resp. $\mathcal{C^{\prime\prime}}$ ) and by </p>
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0003042v1
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[{"type": "text", "text": "Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\\mathrm{{S^{3}}}$ (see [5]). ", "page_idx": 2}, {"type": "text", "text": "A set of mutually disjoint arcs $\\{t_{1},\\ldots,t_{n}\\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ such that $t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}$ , $t_{i}\\cap D_{j}=\\emptyset$ and $\\partial D_{i}-t_{i}\\subset\\partial U$ for $1\\leq i,j\\leq n$ and $i\\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot. ", "page_idx": 2}, {"type": "text", "text": "The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\\prime}$ is defined only for $g\\geq g^{\\prime}$ and the genus 0 bridge number of a link in $\\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\\geq0$ . ", "page_idx": 2}, {"type": "text", "text": "In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23]. ", "page_idx": 2}, {"type": "text", "text": "2 Dunwoody manifolds ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\\ >\\ 0$ , $a,b,c\\,\\geq\\,0$ and $a+b+c>0$ . Let $\\Gamma=\\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1. ", "page_idx": 2}, {"type": "text", "text": "It contains $n$ upper cycles $C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}$ and $n$ lower cycles $C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\\dots,n$ , the cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) is connected to the cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\\prime\\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\\prime\\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}$ and $\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}$ . Moreover, denote by $A^{\\prime}$ (resp. $A^{\\prime\\prime}$ ) the set of the arcs of $\\Gamma$ belonging to a cycle of $\\mathcal{C}^{\\prime}$ (resp. $\\mathcal{C^{\\prime\\prime}}$ ) and by ", "page_idx": 2}]
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1.0, "text": ""}, {"category_id": 15, "poly": [382.0, 1379.0, 884.0, 1379.0, 884.0, 1428.0, 382.0, 1428.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [835.0, 1924.0, 861.0, 1924.0, 861.0, 1959.0, 835.0, 1959.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 2, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [109, 125, 500, 168], "lines": [{"bbox": [127, 127, 498, 142], "spans": [{"bbox": [127, 127, 367, 142], "score": 1.0, "content": "Following [10], we recall the definition of genus ", "type": "text"}, {"bbox": [368, 133, 374, 141], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [374, 127, 498, 142], "score": 1.0, "content": " bridge number of a link,", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 500, 156], "score": 1.0, "content": "which is a generalization of the classical concept of bridge number for links", "type": "text"}], "index": 1}, {"bbox": [109, 154, 186, 171], "spans": [{"bbox": [109, 154, 123, 171], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 157, 137, 167], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 154, 186, 171], "score": 1.0, "content": " (see [5]).", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 169, 500, 298], "lines": [{"bbox": [126, 170, 501, 187], "spans": [{"bbox": [126, 170, 303, 187], "score": 1.0, "content": "A set of mutually disjoint arcs ", "type": "text"}, {"bbox": [303, 172, 363, 185], "score": 0.94, "content": "\\{t_{1},\\ldots,t_{n}\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [363, 170, 501, 187], "score": 1.0, "content": " properly embedded in a", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 175, 199], "score": 1.0, "content": "handlebody ", "type": "text"}, {"bbox": [176, 187, 185, 196], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [185, 185, 472, 199], "score": 1.0, "content": " is trivial if there is a set of mutually disjoint discs ", "type": "text"}, {"bbox": [473, 186, 500, 198], "score": 0.84, "content": "D\\,=", "type": "inline_equation", "height": 12, "width": 27}], "index": 4}, {"bbox": [110, 200, 499, 214], "spans": [{"bbox": [110, 201, 180, 213], "score": 0.94, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 200, 233, 214], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [234, 201, 342, 212], "score": 0.9, "content": "t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}", "type": "inline_equation", "height": 11, "width": 108}, {"bbox": [343, 200, 348, 214], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [348, 201, 402, 214], "score": 0.94, "content": "t_{i}\\cap D_{j}=\\emptyset", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 200, 427, 214], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [428, 201, 499, 212], "score": 0.92, "content": "\\partial D_{i}-t_{i}\\subset\\partial U", "type": "inline_equation", "height": 11, "width": 71}], "index": 5}, {"bbox": [109, 214, 500, 228], "spans": [{"bbox": [109, 214, 128, 228], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [128, 216, 188, 227], "score": 0.94, "content": "1\\leq i,j\\leq n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [188, 214, 213, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 216, 239, 227], "score": 0.93, "content": "i\\neq j", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [240, 214, 268, 228], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [268, 216, 281, 226], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [282, 214, 307, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [308, 216, 321, 226], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [321, 214, 500, 228], "score": 1.0, "content": " be the two handlebodies of a Hee-", "type": "text"}], "index": 6}, {"bbox": [109, 230, 500, 242], "spans": [{"bbox": [109, 230, 381, 242], "score": 1.0, "content": "gaard splitting of the closed orientable 3-manifold ", "type": "text"}, {"bbox": [381, 231, 394, 240], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [395, 230, 442, 242], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [443, 231, 452, 239], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [452, 230, 500, 242], "score": 1.0, "content": " be their", "type": "text"}], "index": 7}, {"bbox": [109, 244, 501, 257], "spans": [{"bbox": [109, 244, 235, 257], "score": 1.0, "content": "common surface: a link ", "type": "text"}, {"bbox": [236, 245, 244, 254], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [244, 244, 262, 257], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [262, 245, 275, 254], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [275, 244, 304, 257], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [305, 248, 312, 254], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [313, 244, 478, 257], "score": 1.0, "content": "-bridge position with respect to ", "type": "text"}, {"bbox": [479, 245, 488, 254], "score": 0.91, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 244, 501, 257], "score": 1.0, "content": " if", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 271], "spans": [{"bbox": [110, 259, 118, 268], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [119, 257, 174, 271], "score": 1.0, "content": " intersects ", "type": "text"}, {"bbox": [174, 259, 183, 268], "score": 0.92, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 257, 368, 271], "score": 1.0, "content": " transversally and if the set of arcs ", "type": "text"}, {"bbox": [368, 259, 402, 270], "score": 0.94, "content": "L\\cap U_{i}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 257, 426, 271], "score": 1.0, "content": " has ", "type": "text"}, {"bbox": [426, 263, 434, 268], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [434, 257, 500, 271], "score": 1.0, "content": " components", "type": "text"}], "index": 9}, {"bbox": [109, 271, 499, 286], "spans": [{"bbox": [109, 271, 222, 286], "score": 1.0, "content": "and is trivial both in ", "type": "text"}, {"bbox": [222, 274, 235, 284], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [235, 271, 276, 286], "score": 1.0, "content": " and in ", "type": "text"}, {"bbox": [276, 274, 289, 284], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [289, 271, 499, 286], "score": 1.0, "content": ". A link in 1-bridge position is obviously", "type": "text"}], "index": 10}, {"bbox": [109, 287, 147, 299], "spans": [{"bbox": [109, 287, 147, 299], "score": 1.0, "content": "a knot.", "type": "text"}], "index": 11}], "index": 7}, {"type": "text", "bbox": [109, 299, 500, 399], "lines": [{"bbox": [127, 300, 500, 315], "spans": [{"bbox": [127, 300, 182, 315], "score": 1.0, "content": "The genus ", "type": "text"}, {"bbox": [182, 306, 189, 314], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [189, 300, 312, 315], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [313, 303, 321, 312], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [321, 300, 336, 315], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [337, 303, 349, 312], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 300, 356, 315], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [356, 302, 383, 315], "score": 0.95, "content": "b_{g}(L)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [383, 300, 500, 315], "score": 1.0, "content": ", is the smallest integer", "type": "text"}], "index": 12}, {"bbox": [110, 315, 501, 330], "spans": [{"bbox": [110, 320, 117, 326], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 315, 172, 330], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [172, 317, 180, 326], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [180, 315, 208, 330], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [208, 320, 216, 326], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [216, 315, 441, 330], "score": 1.0, "content": "-bridge position with respect to some genus ", "type": "text"}, {"bbox": [441, 320, 447, 328], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [448, 315, 501, 330], "score": 1.0, "content": " Heegaard", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 163, 344], "score": 1.0, "content": "surface in ", "type": "text"}, {"bbox": [163, 331, 176, 340], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [176, 330, 247, 344], "score": 1.0, "content": ". If the genus ", "type": "text"}, {"bbox": [248, 335, 254, 343], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [254, 330, 381, 344], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [390, 330, 404, 344], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [405, 331, 410, 340], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [410, 330, 479, 344], "score": 1.0, "content": ", we say that ", "type": "text"}, {"bbox": [479, 331, 487, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [487, 330, 500, 344], "score": 1.0, "content": " is", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 152, 358], "score": 1.0, "content": "a genus ", "type": "text"}, {"bbox": [152, 349, 158, 357], "score": 0.85, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [159, 345, 162, 358], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [162, 346, 167, 355], "score": 0.84, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [168, 345, 291, 358], "score": 1.0, "content": "-bridge link or simply a ", "type": "text"}, {"bbox": [292, 345, 317, 358], "score": 0.94, "content": "(g,b)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [317, 345, 457, 358], "score": 1.0, "content": "-link. Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [480, 597, 494, 609], "score": 0.87, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [494, 595, 500, 611], "score": 1.0, "content": ")", "type": "text"}], "index": 29}, {"bbox": [108, 609, 499, 625], "spans": [{"bbox": [108, 609, 237, 625], "score": 1.0, "content": "is connected to the cycle ", "type": "text"}, {"bbox": [238, 612, 261, 624], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [261, 609, 298, 625], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [298, 612, 321, 624], "score": 0.91, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [321, 609, 344, 625], "score": 1.0, "content": ") by ", "type": "text"}, {"bbox": [344, 615, 351, 621], "score": 0.87, "content": "a", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [351, 609, 484, 625], "score": 1.0, "content": " parallel arcs, to the cycle ", "type": "text"}, {"bbox": [484, 612, 499, 624], "score": 0.91, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}], "index": 30}, {"bbox": [107, 623, 501, 643], "spans": [{"bbox": [107, 623, 126, 643], "score": 1.0, "content": "by ", "type": "text"}, {"bbox": [126, 630, 131, 635], "score": 0.87, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [132, 623, 284, 643], "score": 1.0, "content": " parallel arcs and to the cycle ", "type": "text"}, {"bbox": [284, 626, 307, 639], "score": 0.93, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [307, 623, 325, 643], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [326, 626, 331, 635], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [331, 623, 443, 643], "score": 1.0, "content": " parallel arcs (assume ", "type": "text"}, {"bbox": [444, 627, 491, 636], "score": 0.91, "content": "n+1=1", "type": "inline_equation", "height": 9, "width": 47}, {"bbox": [492, 623, 501, 643], "score": 1.0, "content": ").", "type": "text"}], "index": 31}, {"bbox": [108, 637, 499, 655], "spans": [{"bbox": [108, 637, 149, 655], "score": 1.0, "content": "We set ", "type": "text"}, {"bbox": [149, 641, 243, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}", "type": "inline_equation", "height": 12, "width": 94}, {"bbox": [244, 637, 270, 655], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [270, 640, 368, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [369, 637, 487, 655], "score": 1.0, "content": ". Moreover, denote by ", "type": "text"}, {"bbox": [487, 641, 499, 650], "score": 0.89, "content": "A^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}], "index": 32}, {"bbox": [110, 653, 499, 668], "spans": [{"bbox": [110, 653, 144, 668], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [144, 655, 158, 664], "score": 0.84, "content": "A^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [158, 653, 271, 668], "score": 1.0, "content": ") the set of the arcs of ", "type": "text"}, {"bbox": [271, 656, 278, 664], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [279, 653, 397, 668], "score": 1.0, "content": " belonging to a cycle of ", "type": "text"}, {"bbox": [397, 655, 407, 664], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [408, 653, 444, 668], "score": 1.0, "content": " (resp. 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Let ", "type": "text"}, {"bbox": [268, 216, 281, 226], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [282, 214, 307, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [308, 216, 321, 226], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [321, 214, 500, 228], "score": 1.0, "content": " be the two handlebodies of a Hee-", "type": "text"}], "index": 6}, {"bbox": [109, 230, 500, 242], "spans": [{"bbox": [109, 230, 381, 242], "score": 1.0, "content": "gaard splitting of the closed orientable 3-manifold ", "type": "text"}, {"bbox": [381, 231, 394, 240], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [395, 230, 442, 242], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [443, 231, 452, 239], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [452, 230, 500, 242], "score": 1.0, "content": " be their", "type": "text"}], "index": 7}, {"bbox": [109, 244, 501, 257], "spans": [{"bbox": [109, 244, 235, 257], "score": 1.0, "content": "common surface: a link ", "type": "text"}, {"bbox": [236, 245, 244, 254], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [244, 244, 262, 257], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [262, 245, 275, 254], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [275, 244, 304, 257], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [305, 248, 312, 254], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [313, 244, 478, 257], "score": 1.0, "content": "-bridge position with respect to ", "type": "text"}, {"bbox": [479, 245, 488, 254], "score": 0.91, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 244, 501, 257], "score": 1.0, "content": " if", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 271], "spans": [{"bbox": [110, 259, 118, 268], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [119, 257, 174, 271], "score": 1.0, "content": " intersects ", "type": "text"}, {"bbox": [174, 259, 183, 268], "score": 0.92, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 257, 368, 271], "score": 1.0, "content": " transversally and if the set of arcs ", "type": "text"}, {"bbox": [368, 259, 402, 270], "score": 0.94, "content": "L\\cap U_{i}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 257, 426, 271], "score": 1.0, "content": " has ", "type": "text"}, {"bbox": [426, 263, 434, 268], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [434, 257, 500, 271], "score": 1.0, "content": " components", "type": "text"}], "index": 9}, {"bbox": [109, 271, 499, 286], "spans": [{"bbox": [109, 271, 222, 286], "score": 1.0, "content": "and is trivial both in ", "type": "text"}, {"bbox": [222, 274, 235, 284], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [235, 271, 276, 286], "score": 1.0, "content": " and in ", "type": "text"}, {"bbox": [276, 274, 289, 284], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [289, 271, 499, 286], "score": 1.0, "content": ". A link in 1-bridge position is obviously", "type": "text"}], "index": 10}, {"bbox": [109, 287, 147, 299], "spans": [{"bbox": [109, 287, 147, 299], "score": 1.0, "content": "a knot.", "type": "text"}], "index": 11}], "index": 7, "bbox_fs": [109, 170, 501, 299]}, {"type": "text", "bbox": [109, 299, 500, 399], "lines": [{"bbox": [127, 300, 500, 315], "spans": [{"bbox": [127, 300, 182, 315], "score": 1.0, "content": "The genus ", "type": "text"}, {"bbox": [182, 306, 189, 314], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [189, 300, 312, 315], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [313, 303, 321, 312], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [321, 300, 336, 315], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [337, 303, 349, 312], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 300, 356, 315], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [356, 302, 383, 315], "score": 0.95, "content": "b_{g}(L)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [383, 300, 500, 315], "score": 1.0, "content": ", is the smallest integer", "type": "text"}], "index": 12}, {"bbox": [110, 315, 501, 330], "spans": [{"bbox": [110, 320, 117, 326], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 315, 172, 330], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [172, 317, 180, 326], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [180, 315, 208, 330], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [208, 320, 216, 326], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [216, 315, 441, 330], "score": 1.0, "content": "-bridge position with respect to some genus ", "type": "text"}, {"bbox": [441, 320, 447, 328], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [448, 315, 501, 330], "score": 1.0, "content": " Heegaard", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 163, 344], "score": 1.0, "content": "surface in ", "type": "text"}, {"bbox": [163, 331, 176, 340], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [176, 330, 247, 344], "score": 1.0, "content": ". If the genus ", "type": "text"}, {"bbox": [248, 335, 254, 343], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [254, 330, 381, 344], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [390, 330, 404, 344], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [405, 331, 410, 340], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [410, 330, 479, 344], "score": 1.0, "content": ", we say that ", "type": "text"}, {"bbox": [479, 331, 487, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [487, 330, 500, 344], "score": 1.0, "content": " is", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 152, 358], "score": 1.0, "content": "a genus ", "type": "text"}, {"bbox": [152, 349, 158, 357], "score": 0.85, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [159, 345, 162, 358], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [162, 346, 167, 355], "score": 0.84, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [168, 345, 291, 358], "score": 1.0, "content": "-bridge link or simply a ", "type": "text"}, {"bbox": [292, 345, 317, 358], "score": 0.94, "content": "(g,b)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [317, 345, 457, 358], "score": 1.0, "content": "-link. Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15, "bbox_fs": [109, 300, 501, 401]}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21, "bbox_fs": [109, 402, 501, 475]}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [109, 525, 500, 569]}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [480, 597, 494, 609], "score": 0.87, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [494, 595, 500, 611], "score": 1.0, "content": ")", "type": "text"}], "index": 29}, {"bbox": [108, 609, 499, 625], "spans": [{"bbox": [108, 609, 237, 625], "score": 1.0, "content": "is connected to the cycle ", "type": "text"}, {"bbox": [238, 612, 261, 624], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [261, 609, 298, 625], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [298, 612, 321, 624], "score": 0.91, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [321, 609, 344, 625], "score": 1.0, "content": ") by ", "type": "text"}, {"bbox": [344, 615, 351, 621], "score": 0.87, "content": "a", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [351, 609, 484, 625], "score": 1.0, "content": " parallel arcs, to the cycle ", "type": "text"}, {"bbox": [484, 612, 499, 624], "score": 0.91, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}], "index": 30}, {"bbox": [107, 623, 501, 643], "spans": [{"bbox": [107, 623, 126, 643], "score": 1.0, "content": "by ", "type": "text"}, {"bbox": [126, 630, 131, 635], "score": 0.87, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [132, 623, 284, 643], "score": 1.0, "content": " parallel arcs and to the cycle ", "type": "text"}, {"bbox": [284, 626, 307, 639], "score": 0.93, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [307, 623, 325, 643], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [326, 626, 331, 635], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [331, 623, 443, 643], "score": 1.0, "content": " parallel arcs (assume ", "type": "text"}, {"bbox": [444, 627, 491, 636], "score": 0.91, "content": "n+1=1", "type": "inline_equation", "height": 9, "width": 47}, {"bbox": [492, 623, 501, 643], "score": 1.0, "content": ").", "type": "text"}], "index": 31}, {"bbox": [108, 637, 499, 655], "spans": [{"bbox": [108, 637, 149, 655], "score": 1.0, "content": "We set ", "type": "text"}, {"bbox": [149, 641, 243, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}", "type": "inline_equation", "height": 12, "width": 94}, {"bbox": [244, 637, 270, 655], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [270, 640, 368, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [369, 637, 487, 655], "score": 1.0, "content": ". Moreover, denote by ", "type": "text"}, {"bbox": [487, 641, 499, 650], "score": 0.89, "content": "A^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}], "index": 32}, {"bbox": [110, 653, 499, 668], "spans": [{"bbox": [110, 653, 144, 668], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [144, 655, 158, 664], "score": 0.84, "content": "A^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [158, 653, 271, 668], "score": 1.0, "content": ") the set of the arcs of ", "type": "text"}, {"bbox": [271, 656, 278, 664], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [279, 653, 397, 668], "score": 1.0, "content": " belonging to a cycle of ", "type": "text"}, {"bbox": [397, 655, 407, 664], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [408, 653, 444, 668], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [444, 655, 457, 664], "score": 0.86, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [457, 653, 499, 668], "score": 1.0, "content": ") and by", "type": "text"}], "index": 33}], "index": 30.5, "bbox_fs": [107, 578, 503, 668]}]}
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[{"type": "text", "bbox": [109, 125, 500, 168], "content": "Following [10], we recall the definition of genus bridge number of a link, which is a generalization of the classical concept of bridge number for links in (see [5]).", "index": 0}, {"type": "text", "bbox": [109, 169, 500, 298], "content": "A set of mutually disjoint arcs properly embedded in a handlebody is trivial if there is a set of mutually disjoint discs such that , and for and . Let and be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold and let be their common surface: a link in is in -bridge position with respect to if intersects transversally and if the set of arcs has components and is trivial both in and in . A link in 1-bridge position is obviously a knot.", "index": 1}, {"type": "text", "bbox": [109, 299, 500, 399], "content": "The genus bridge number of a link in , , is the smallest integer for which is in -bridge position with respect to some genus Heegaard surface in . If the genus bridge number of a link is , we say that is a genus -bridge link or simply a -link. Of course, the genus bridge number of a link in a manifold of Heegaard genus is defined only for and the genus 0 bridge number of a link in is the classical bridge number. Moreover, a -link is a knot, for each .", "index": 2}, {"type": "text", "bbox": [109, 400, 500, 473], "content": "In what follows, we shall deal with -knots, i.e. knots in or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of -knots in contains all torus knots (trivially) and all 2-bridge knots (i.e. -knots) [23].", "index": 3}, {"type": "title", "bbox": [110, 493, 318, 513], "content": "2 Dunwoody manifolds", "index": 4}, {"type": "text", "bbox": [109, 524, 500, 567], "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let be integers such that , and . Let be the planar regular trivalent graph drawn in Figure 1.", "index": 5}, {"type": "text", "bbox": [110, 579, 500, 667], "content": "It contains upper cycles and lower cycles , each having vertices. For each , the cycle (resp. ) is connected to the cycle (resp. ) by parallel arcs, to the cycle by parallel arcs and to the cycle by parallel arcs (assume ). We set and . Moreover, denote by (resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by", "index": 6}]
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[{"bbox": [127, 127, 498, 142], "content": "Following [10], we recall the definition of genus bridge number of a link,", "parent_index": 0, "line_index": 0}, {"bbox": [109, 142, 500, 156], "content": "which is a generalization of the classical concept of bridge number for links", "parent_index": 0, "line_index": 1}, {"bbox": [109, 154, 186, 171], "content": "in (see [5]).", "parent_index": 0, "line_index": 2}, {"bbox": [126, 170, 501, 187], "content": "A set of mutually disjoint arcs properly embedded in a", "parent_index": 1, "line_index": 0}, {"bbox": [109, 185, 500, 199], "content": "handlebody is trivial if there is a set of mutually disjoint discs", "parent_index": 1, "line_index": 1}, {"bbox": [110, 200, 499, 214], "content": "such that , and", "parent_index": 1, "line_index": 2}, {"bbox": [109, 214, 500, 228], "content": "for and . Let and be the two handlebodies of a Hee-", "parent_index": 1, "line_index": 3}, {"bbox": [109, 230, 500, 242], "content": "gaard splitting of the closed orientable 3-manifold and let be their", "parent_index": 1, "line_index": 4}, {"bbox": [109, 244, 501, 257], "content": "common surface: a link in is in -bridge position with respect to if", "parent_index": 1, "line_index": 5}, {"bbox": [110, 257, 500, 271], "content": "intersects transversally and if the set of arcs has components", "parent_index": 1, "line_index": 6}, {"bbox": [109, 271, 499, 286], "content": "and is trivial both in and in . A link in 1-bridge position is obviously", "parent_index": 1, "line_index": 7}, {"bbox": [109, 287, 147, 299], "content": "a knot.", "parent_index": 1, "line_index": 8}, {"bbox": [127, 300, 500, 315], "content": "The genus bridge number of a link in , , is the smallest integer", "parent_index": 2, "line_index": 0}, {"bbox": [110, 315, 501, 330], "content": "for which is in -bridge position with respect to some genus Heegaard", "parent_index": 2, "line_index": 1}, {"bbox": [110, 330, 500, 344], "content": "surface in . If the genus bridge number of a link is , we say that is", "parent_index": 2, "line_index": 2}, {"bbox": [110, 345, 500, 358], "content": "a genus -bridge link or simply a -link. Of course, the genus bridge", "parent_index": 2, "line_index": 3}, {"bbox": [109, 358, 499, 374], "content": "number of a link in a manifold of Heegaard genus is defined only for", "parent_index": 2, "line_index": 4}, {"bbox": [110, 373, 499, 387], "content": "and the genus 0 bridge number of a link in is the classical bridge number.", "parent_index": 2, "line_index": 5}, {"bbox": [110, 387, 356, 401], "content": "Moreover, a -link is a knot, for each .", "parent_index": 2, "line_index": 6}, {"bbox": [126, 402, 501, 416], "content": "In what follows, we shall deal with -knots, i.e. knots in or in lens", "parent_index": 3, "line_index": 0}, {"bbox": [109, 416, 501, 430], "content": "spaces. This class of knots is very important in the light of some results and", "parent_index": 3, "line_index": 1}, {"bbox": [110, 431, 499, 446], "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "parent_index": 3, "line_index": 2}, {"bbox": [109, 444, 499, 460], "content": "Notice that the class of -knots in contains all torus knots (trivially)", "parent_index": 3, "line_index": 3}, {"bbox": [110, 459, 340, 475], "content": "and all 2-bridge knots (i.e. -knots) [23].", "parent_index": 3, "line_index": 4}, {"bbox": [110, 496, 318, 514], "content": "2 Dunwoody manifolds", "parent_index": 4, "line_index": 0}, {"bbox": [109, 525, 499, 541], "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "parent_index": 5, "line_index": 0}, {"bbox": [110, 541, 500, 554], "content": "be integers such that , and . Let", "parent_index": 5, "line_index": 1}, {"bbox": [110, 555, 481, 569], "content": "be the planar regular trivalent graph drawn in Figure 1.", "parent_index": 5, "line_index": 2}, {"bbox": [123, 578, 503, 599], "content": "It contains upper cycles and lower cycles , each", "parent_index": 6, "line_index": 0}, {"bbox": [109, 595, 500, 611], "content": "having vertices. For each , the cycle (resp. )", "parent_index": 6, "line_index": 1}, {"bbox": [108, 609, 499, 625], "content": "is connected to the cycle (resp. ) by parallel arcs, to the cycle", "parent_index": 6, "line_index": 2}, {"bbox": [107, 623, 501, 643], "content": "by parallel arcs and to the cycle by parallel arcs (assume ).", "parent_index": 6, "line_index": 3}, {"bbox": [108, 637, 499, 655], "content": "We set and . Moreover, denote by", "parent_index": 6, "line_index": 4}, {"bbox": [110, 653, 499, 668], "content": "(resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by", "parent_index": 6, "line_index": 5}]
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[]
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[{"bbox": [368, 133, 374, 141], "content": "g", "parent_index": 0, "subtype": "inline"}, {"bbox": [124, 157, 137, 167], "content": "\\mathrm{{S^{3}}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [303, 172, 363, 185], "content": "\\{t_{1},\\ldots,t_{n}\\}", "parent_index": 1, "subtype": "inline"}, {"bbox": [176, 187, 185, 196], "content": "U", "parent_index": 1, "subtype": "inline"}, {"bbox": [473, 186, 500, 198], "content": "D\\,=", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 201, 180, 213], "content": "\\{D_{1},...\\,,D_{n}\\}", "parent_index": 1, "subtype": "inline"}, {"bbox": [234, 201, 342, 212], "content": "t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}", "parent_index": 1, "subtype": "inline"}, {"bbox": [348, 201, 402, 214], "content": "t_{i}\\cap D_{j}=\\emptyset", "parent_index": 1, "subtype": "inline"}, {"bbox": [428, 201, 499, 212], "content": "\\partial D_{i}-t_{i}\\subset\\partial U", "parent_index": 1, "subtype": "inline"}, {"bbox": [128, 216, 188, 227], "content": "1\\leq i,j\\leq n", "parent_index": 1, "subtype": "inline"}, {"bbox": [214, 216, 239, 227], "content": "i\\neq j", "parent_index": 1, "subtype": "inline"}, {"bbox": [268, 216, 281, 226], "content": "U_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [308, 216, 321, 226], "content": "U_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [381, 231, 394, 240], "content": "M", "parent_index": 1, "subtype": "inline"}, {"bbox": [443, 231, 452, 239], "content": "T", "parent_index": 1, "subtype": "inline"}, {"bbox": [236, 245, 244, 254], "content": "L", "parent_index": 1, "subtype": "inline"}, {"bbox": [262, 245, 275, 254], "content": "M", "parent_index": 1, "subtype": "inline"}, {"bbox": [305, 248, 312, 254], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [479, 245, 488, 254], "content": "T", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 259, 118, 268], "content": "L", "parent_index": 1, "subtype": "inline"}, {"bbox": [174, 259, 183, 268], "content": "T", "parent_index": 1, "subtype": "inline"}, {"bbox": [368, 259, 402, 270], "content": "L\\cap U_{i}", "parent_index": 1, "subtype": "inline"}, {"bbox": [426, 263, 434, 268], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [222, 274, 235, 284], "content": "U_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [276, 274, 289, 284], "content": "U_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [182, 306, 189, 314], "content": "g", "parent_index": 2, "subtype": "inline"}, {"bbox": [313, 303, 321, 312], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [337, 303, 349, 312], "content": "M", "parent_index": 2, "subtype": "inline"}, {"bbox": [356, 302, 383, 315], "content": "b_{g}(L)", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 320, 117, 326], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [172, 317, 180, 326], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [208, 320, 216, 326], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [441, 320, 447, 328], "content": "g", "parent_index": 2, "subtype": "inline"}, {"bbox": [163, 331, 176, 340], "content": "M", "parent_index": 2, "subtype": "inline"}, {"bbox": [248, 335, 254, 343], "content": "g", "parent_index": 2, "subtype": "inline"}, {"bbox": [381, 331, 389, 340], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 331, 410, 340], "content": "b", "parent_index": 2, "subtype": "inline"}, {"bbox": [479, 331, 487, 340], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 349, 158, 357], "content": "g", "parent_index": 2, "subtype": "inline"}, {"bbox": [162, 346, 167, 355], "content": "b", "parent_index": 2, "subtype": "inline"}, {"bbox": [292, 345, 317, 358], "content": "(g,b)", "parent_index": 2, "subtype": "inline"}, {"bbox": [457, 349, 463, 357], "content": "g", "parent_index": 2, "subtype": "inline"}, {"bbox": [362, 360, 371, 372], "content": "g^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [468, 360, 499, 372], "content": "g\\geq g^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [329, 374, 342, 384], "content": "\\mathbf{S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [174, 389, 201, 401], "content": "(g,1)", "parent_index": 2, "subtype": "inline"}, {"bbox": [324, 390, 352, 401], "content": "g\\geq0", "parent_index": 2, "subtype": "inline"}, {"bbox": [307, 403, 333, 416], "content": "(1,1)", "parent_index": 3, "subtype": "inline"}, {"bbox": [435, 403, 448, 413], "content": "\\mathrm{{S^{3}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [234, 446, 259, 459], "content": "(1,1)", "parent_index": 3, "subtype": "inline"}, {"bbox": [309, 446, 321, 456], "content": "\\mathbf{S^{3}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [251, 461, 277, 474], "content": "(0,2)", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 542, 149, 554], "content": "a,b,c,n", "parent_index": 5, "subtype": "inline"}, {"bbox": [270, 543, 304, 551], "content": "n\\ >\\ 0", "parent_index": 5, "subtype": "inline"}, {"bbox": [312, 542, 366, 554], "content": "a,b,c\\,\\geq\\,0", "parent_index": 5, "subtype": "inline"}, {"bbox": [394, 542, 469, 552], "content": "a+b+c>0", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 556, 189, 568], "content": "\\Gamma=\\Gamma(a,b,c,n)", "parent_index": 5, "subtype": "inline"}, {"bbox": [185, 586, 192, 592], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [260, 583, 316, 595], "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [340, 586, 347, 592], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [413, 583, 470, 595], "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [147, 597, 218, 607], "content": "d=2a+b+c", "parent_index": 6, "subtype": "inline"}, {"bbox": [315, 598, 375, 609], "content": "i=1,\\dots,n", "parent_index": 6, "subtype": "inline"}, {"bbox": [430, 597, 443, 609], "content": "C_{i}^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [480, 597, 494, 609], "content": "C_{i}^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [238, 612, 261, 624], "content": "C_{i+1}^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [298, 612, 321, 624], "content": "C_{i+1}^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [344, 615, 351, 621], "content": "a", "parent_index": 6, "subtype": "inline"}, {"bbox": [484, 612, 499, 624], "content": "C_{i}^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [126, 630, 131, 635], "content": "c", "parent_index": 6, "subtype": "inline"}, {"bbox": [284, 626, 307, 639], "content": "C_{i+1}^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [326, 626, 331, 635], "content": "b", "parent_index": 6, "subtype": "inline"}, {"bbox": [444, 627, 491, 636], "content": "n+1=1", "parent_index": 6, "subtype": "inline"}, {"bbox": [149, 641, 243, 653], "content": "\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [270, 640, 368, 653], "content": "\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [487, 641, 499, 650], "content": "A^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [144, 655, 158, 664], "content": "A^{\\prime\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [271, 656, 278, 664], "content": "\\Gamma", "parent_index": 6, "subtype": "inline"}, {"bbox": [397, 655, 407, 664], "content": "\\mathcal{C}^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [444, 655, 457, 664], "content": "\\mathcal{C^{\\prime\\prime}}", "parent_index": 6, "subtype": "inline"}]
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[]
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Figure 1: The graph $\Gamma$ .
$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\Gamma$ in $\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\rho_{n}$ of the sphere by $2\pi/n$ radians along a suitable axis intersecting $\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\rho_{n}$ sends $C_{i}^{\prime}$ to $C_{i+1}^{\prime}$ and $C_{i}^{\prime\prime}$ to $C_{i+1}^{\prime\prime}$ (mod $n$ ), for each $i=1,\dots,n$ .
By cutting the sphere along all $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\mathcal{C}^{\prime}$ (resp. of $\mathcal{C^{\prime\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that:
- the vertex 1 of each $C_{i}^{\prime}$ is the endpoint of the first arc of $A$ connecting Ci′ with Ci′+1;
- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\prime\prime}$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.
Then glue the cycle $C_{i}^{\prime}$ with the cycle $C_{i-s}^{\prime\prime}$ (mod $n$ ) so that equally labelled vertices are identified together.
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<html><body>
<div class="image" data-bbox="155 120 454 328"><img data-bbox="155 120 454 328"/><p class="caption" data-bbox="244 349 364 364">Figure 1: The graph $\Gamma$ . </p></div>
<p data-bbox="109 383 500 470">$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\Gamma$ in $\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\rho_{n}$ of the sphere by $2\pi/n$ radians along a suitable axis intersecting $\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\rho_{n}$ sends $C_{i}^{\prime}$ to $C_{i+1}^{\prime}$ and $C_{i}^{\prime\prime}$ to $C_{i+1}^{\prime\prime}$ (mod $n$ ), for each $i=1,\dots,n$ . </p>
<p data-bbox="110 470 501 543">By cutting the sphere along all $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\mathcal{C}^{\prime}$ (resp. of $\mathcal{C^{\prime\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that: </p>
<p data-bbox="126 551 502 622">- the vertex 1 of each $C_{i}^{\prime}$ is the endpoint of the first arc of $A$ connecting Ci′ with Ci′+1;
- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\prime\prime}$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1. </p>
<p data-bbox="109 630 500 659">Then glue the cycle $C_{i}^{\prime}$ with the cycle $C_{i-s}^{\prime\prime}$ (mod $n$ ) so that equally labelled vertices are identified together. </p>
</body></html>
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0003042v1
| 3 | 612 | 792 | 1,275 | 1,650 |
[{"type": "image", "img_path": "images/75b9851aa6f587d46bce7b58819bdbd6615243f7ae4ed18bcd8b379ea5f40807.jpg", "img_caption": ["Figure 1: The graph $\\Gamma$ . "], "img_footnote": [], "page_idx": 3}, {"type": "text", "text": "$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\\Gamma$ in $\\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\\rho_{n}$ of the sphere by $2\\pi/n$ radians along a suitable axis intersecting $\\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\\rho_{n}$ sends $C_{i}^{\\prime}$ to $C_{i+1}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ to $C_{i+1}^{\\prime\\prime}$ (mod $n$ ), for each $i=1,\\dots,n$ . ", "page_idx": 3}, {"type": "text", "text": "By cutting the sphere along all $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\\mathcal{C}^{\\prime}$ (resp. of $\\mathcal{C^{\\prime\\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that: ", "page_idx": 3}, {"type": "text", "text": "- the vertex 1 of each $C_{i}^{\\prime}$ is the endpoint of the first arc of $A$ connecting Ci′ with Ci′+1; \n- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\\prime\\prime}$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1. ", "page_idx": 3}, {"type": "text", "text": "Then glue the cycle $C_{i}^{\\prime}$ with the cycle $C_{i-s}^{\\prime\\prime}$ (mod $n$ ) so that equally labelled vertices are identified together. ", "page_idx": 3}]
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The one-point compactification of", "type": "text"}], "index": 15}, {"bbox": [110, 400, 501, 415], "spans": [{"bbox": [110, 400, 369, 415], "score": 1.0, "content": "the plane leads to a 2-cell embedding of the graph ", "type": "text"}, {"bbox": [369, 402, 377, 411], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [377, 400, 393, 415], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [394, 401, 407, 411], "score": 0.9, "content": "\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [407, 400, 501, 415], "score": 1.0, "content": "; it is evident that", "type": "text"}], "index": 16}, {"bbox": [109, 414, 499, 430], "spans": [{"bbox": [109, 414, 366, 430], "score": 1.0, "content": "the graph is invariant with respect to a rotation ", "type": "text"}, {"bbox": [367, 420, 379, 428], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [379, 414, 473, 430], "score": 1.0, "content": " of the sphere by ", "type": "text"}, {"bbox": [473, 416, 499, 428], "score": 0.94, "content": "2\\pi/n", "type": "inline_equation", "height": 12, "width": 26}], "index": 17}, {"bbox": [109, 429, 501, 443], "spans": [{"bbox": [109, 429, 324, 443], "score": 1.0, "content": "radians along a suitable axis intersecting ", "type": "text"}, {"bbox": [324, 430, 338, 440], "score": 0.9, "content": "\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [338, 429, 501, 443], "score": 1.0, "content": " in two points not belonging to", "type": "text"}], "index": 18}, {"bbox": [108, 442, 501, 461], "spans": [{"bbox": [108, 442, 225, 461], "score": 1.0, "content": "the graph. Obviously, ", "type": "text"}, {"bbox": [225, 448, 237, 456], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [237, 442, 271, 461], "score": 1.0, "content": " sends ", "type": "text"}, {"bbox": [272, 445, 284, 457], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [284, 442, 301, 461], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [302, 445, 324, 457], "score": 0.94, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [325, 442, 350, 461], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [351, 445, 365, 457], "score": 0.93, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [365, 442, 383, 461], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [383, 445, 405, 457], "score": 0.93, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [406, 442, 439, 461], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [440, 448, 447, 454], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [447, 442, 501, 461], "score": 1.0, "content": "), for each", "type": "text"}], "index": 19}, {"bbox": [110, 458, 175, 473], "spans": [{"bbox": [110, 460, 171, 471], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [172, 458, 175, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 17.5, "bbox_fs": [108, 385, 501, 473]}, {"type": "text", "bbox": [110, 470, 501, 543], "lines": [{"bbox": [127, 472, 502, 487], "spans": [{"bbox": [127, 472, 286, 487], "score": 1.0, "content": "By cutting the sphere along all ", "type": "text"}, {"bbox": [286, 474, 299, 486], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [299, 472, 324, 487], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [324, 474, 338, 486], "score": 0.93, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [339, 472, 502, 487], "score": 1.0, "content": "and by removing the interior of", "type": "text"}], "index": 21}, {"bbox": [109, 487, 499, 501], "spans": [{"bbox": [109, 487, 364, 501], "score": 1.0, "content": "the corresponding discs, we obtain a sphere with ", "type": "text"}, {"bbox": [364, 489, 377, 497], "score": 0.87, "content": "2n", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [378, 487, 460, 501], "score": 1.0, "content": " holes. Let now ", "type": "text"}, {"bbox": [461, 491, 466, 497], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [467, 487, 493, 501], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 492, 499, 497], "score": 0.87, "content": "s", "type": "inline_equation", "height": 5, "width": 6}], "index": 22}, {"bbox": [109, 501, 501, 516], "spans": [{"bbox": [109, 501, 501, 516], "score": 1.0, "content": "be two new integers; give a clockwise (resp. counterclockwise) orientation to", "type": "text"}], "index": 23}, {"bbox": [110, 516, 500, 529], "spans": [{"bbox": [110, 516, 174, 529], "score": 1.0, "content": "the cycles of ", "type": "text"}, {"bbox": [174, 517, 184, 526], "score": 0.88, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [184, 516, 232, 529], "score": 1.0, "content": " (resp. of ", "type": "text"}, {"bbox": [232, 517, 245, 527], "score": 0.89, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [245, 516, 417, 529], "score": 1.0, "content": ") and label their vertices from 1 to", "type": "text"}, {"bbox": [418, 518, 424, 526], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [425, 516, 500, 529], "score": 1.0, "content": ", in accordance", "type": "text"}], "index": 24}, {"bbox": [110, 530, 345, 544], "spans": [{"bbox": [110, 530, 345, 544], "score": 1.0, "content": "with these orientations (see Figure 2) so that:", "type": "text"}], "index": 25}], "index": 23, "bbox_fs": [109, 472, 502, 544]}, {"type": "list", "bbox": [126, 551, 502, 622], "lines": [{"bbox": [129, 554, 500, 569], "spans": [{"bbox": [129, 554, 244, 569], "score": 1.0, "content": "- the vertex 1 of each ", "type": "text"}, {"bbox": [244, 556, 257, 568], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [257, 554, 431, 569], "score": 1.0, "content": " is the endpoint of the first arc of ", "type": "text"}, {"bbox": [432, 556, 440, 565], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [441, 554, 500, 569], "score": 1.0, "content": " connecting", "type": "text"}], "index": 26, "is_list_start_line": true}, {"bbox": [139, 568, 210, 586], "spans": [{"bbox": [139, 568, 210, 586], "score": 1.0, "content": "Ci′ with Ci′+1;", "type": "text"}], "index": 27, "is_list_end_line": true}, {"bbox": [128, 591, 502, 610], "spans": [{"bbox": [128, 591, 195, 610], "score": 1.0, "content": "- the vertex ", "type": "text"}, {"bbox": [195, 595, 222, 605], "score": 0.91, "content": "1-r", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [222, 591, 257, 610], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [257, 595, 263, 604], "score": 0.8, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [264, 591, 311, 610], "score": 1.0, "content": ") of each ", "type": "text"}, {"bbox": [312, 595, 326, 607], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [326, 591, 502, 610], "score": 1.0, "content": " is the endpoint of the first arc of", "type": "text"}], "index": 28, "is_list_start_line": true}, {"bbox": [137, 602, 285, 625], "spans": [{"bbox": [137, 602, 285, 625], "score": 1.0, "content": "A connecting Ci′′ with Ci′′+1.", "type": "text"}], "index": 29, "is_list_end_line": true}], "index": 27.5, "bbox_fs": [128, 554, 502, 625]}, {"type": "text", "bbox": [109, 630, 500, 659], "lines": [{"bbox": [110, 632, 500, 647], "spans": [{"bbox": [110, 632, 214, 647], "score": 1.0, "content": "Then glue the cycle ", "type": "text"}, {"bbox": [215, 633, 227, 646], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [227, 632, 306, 647], "score": 1.0, "content": " with the cycle ", "type": "text"}, {"bbox": [307, 633, 329, 646], "score": 0.91, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [329, 632, 363, 647], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [363, 637, 371, 642], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [371, 632, 500, 647], "score": 1.0, "content": ") so that equally labelled", "type": "text"}], "index": 30}, {"bbox": [111, 647, 269, 660], "spans": [{"bbox": [111, 647, 269, 660], "score": 1.0, "content": "vertices are identified together.", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [110, 632, 500, 660]}]}
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[{"type": "image", "bbox": [155, 120, 454, 328], "content": "", "index": 0}, {"type": "text", "bbox": [109, 383, 500, 470], "content": "the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph in ; it is evident that the graph is invariant with respect to a rotation of the sphere by radians along a suitable axis intersecting in two points not belonging to the graph. Obviously, sends to and to (mod ), for each .", "index": 1}, {"type": "text", "bbox": [110, 470, 501, 543], "content": "By cutting the sphere along all and and by removing the interior of the corresponding discs, we obtain a sphere with holes. Let now and be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of (resp. of ) and label their vertices from 1 to , in accordance with these orientations (see Figure 2) so that:", "index": 2}, {"type": "list", "bbox": [126, 551, 502, 622], "content": "", "index": 3}, {"type": "text", "bbox": [109, 630, 500, 659], "content": "Then glue the cycle with the cycle (mod ) so that equally labelled vertices are identified together.", "index": 4}]
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[{"bbox": [110, 385, 501, 399], "content": "the set of the other arcs of the graph. The one-point compactification of", "parent_index": 1, "line_index": 0}, {"bbox": [110, 400, 501, 415], "content": "the plane leads to a 2-cell embedding of the graph in ; it is evident that", "parent_index": 1, "line_index": 1}, {"bbox": [109, 414, 499, 430], "content": "the graph is invariant with respect to a rotation of the sphere by", "parent_index": 1, "line_index": 2}, {"bbox": [109, 429, 501, 443], "content": "radians along a suitable axis intersecting in two points not belonging to", "parent_index": 1, "line_index": 3}, {"bbox": [108, 442, 501, 461], "content": "the graph. Obviously, sends to and to (mod ), for each", "parent_index": 1, "line_index": 4}, {"bbox": [110, 458, 175, 473], "content": ".", "parent_index": 1, "line_index": 5}, {"bbox": [127, 472, 502, 487], "content": "By cutting the sphere along all and and by removing the interior of", "parent_index": 2, "line_index": 0}, {"bbox": [109, 487, 499, 501], "content": "the corresponding discs, we obtain a sphere with holes. Let now and", "parent_index": 2, "line_index": 1}, {"bbox": [109, 501, 501, 516], "content": "be two new integers; give a clockwise (resp. counterclockwise) orientation to", "parent_index": 2, "line_index": 2}, {"bbox": [110, 516, 500, 529], "content": "the cycles of (resp. of ) and label their vertices from 1 to , in accordance", "parent_index": 2, "line_index": 3}, {"bbox": [110, 530, 345, 544], "content": "with these orientations (see Figure 2) so that:", "parent_index": 2, "line_index": 4}, {"bbox": [129, 554, 500, 569], "content": "- the vertex 1 of each is the endpoint of the first arc of connecting", "parent_index": 3, "line_index": 0}, {"bbox": [139, 568, 210, 586], "content": "Ci′ with Ci′+1;", "parent_index": 3, "line_index": 1}, {"bbox": [128, 591, 502, 610], "content": "- the vertex (mod ) of each is the endpoint of the first arc of", "parent_index": 3, "line_index": 2}, {"bbox": [137, 602, 285, 625], "content": "A connecting Ci′′ with Ci′′+1.", "parent_index": 3, "line_index": 3}, {"bbox": [110, 632, 500, 647], "content": "Then glue the cycle with the cycle (mod ) so that equally labelled", "parent_index": 4, "line_index": 0}, {"bbox": [111, 647, 269, 660], "content": "vertices are identified together.", "parent_index": 4, "line_index": 1}]
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[{"bbox": [155, 120, 454, 328], "content": "", "parent_index": 0, "subtype": "body"}, {"bbox": [244, 349, 364, 364], "content": "Figure 1: The graph .", "parent_index": 0, "subtype": "caption"}]
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[{"bbox": [110, 387, 119, 396], "content": "A", "parent_index": 1, "subtype": "inline"}, {"bbox": [369, 402, 377, 411], "content": "\\Gamma", "parent_index": 1, "subtype": "inline"}, {"bbox": [394, 401, 407, 411], "content": "\\mathbf{S^{2}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [367, 420, 379, 428], "content": "\\rho_{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [473, 416, 499, 428], "content": "2\\pi/n", "parent_index": 1, "subtype": "inline"}, {"bbox": [324, 430, 338, 440], "content": "\\mathbf{S^{2}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [225, 448, 237, 456], "content": "\\rho_{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [272, 445, 284, 457], "content": "C_{i}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [302, 445, 324, 457], "content": "C_{i+1}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [351, 445, 365, 457], "content": "C_{i}^{\\prime\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [383, 445, 405, 457], "content": "C_{i+1}^{\\prime\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [440, 448, 447, 454], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 460, 171, 471], "content": "i=1,\\dots,n", "parent_index": 1, "subtype": "inline"}, {"bbox": [286, 474, 299, 486], "content": "C_{i}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [324, 474, 338, 486], "content": "C_{i}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [364, 489, 377, 497], "content": "2n", "parent_index": 2, "subtype": "inline"}, {"bbox": [461, 491, 466, 497], "content": "r", "parent_index": 2, "subtype": "inline"}, {"bbox": [493, 492, 499, 497], "content": "s", "parent_index": 2, "subtype": "inline"}, {"bbox": [174, 517, 184, 526], "content": "\\mathcal{C}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [232, 517, 245, 527], "content": "\\mathcal{C^{\\prime\\prime}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [418, 518, 424, 526], "content": "d", "parent_index": 2, "subtype": "inline"}, {"bbox": [244, 556, 257, 568], "content": "C_{i}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [432, 556, 440, 565], "content": "A", "parent_index": 3, "subtype": "inline"}, {"bbox": [195, 595, 222, 605], "content": "1-r", "parent_index": 3, "subtype": "inline"}, {"bbox": [257, 595, 263, 604], "content": "d", "parent_index": 3, "subtype": "inline"}, {"bbox": [312, 595, 326, 607], "content": "C_{i}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [215, 633, 227, 646], "content": "C_{i}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [307, 633, 329, 646], "content": "C_{i-s}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [363, 637, 371, 642], "content": "n", "parent_index": 4, "subtype": "inline"}]
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[]
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Figure 2:
It is evident by construction that the integers $r$ and $s$ can be taken mod $d$ and mod $n$ respectively. Denote by $\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\in\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\geq0$ and $a+b+c>0$ .
The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\mathit{m}}$ cycles $D_{1},\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$ does not disconnect the surface. Set $\mathcal{C}=\{C_{1},\ldots,C_{n}\}$ and $\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$ .
If $m\,=\,n$ and if the cut along the cycles of $\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\scriptscriptstyle\mathcal{C}$ and $\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.
Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\boldsymbol{S}$ satisfying the following conditions:
(1) the set $\mathcal{D}$ contains exactly $n$ cycles;
(2) the surface $T_{n}^{'}$ is not disconnected by the cut along the cycles of $\mathcal{D}$ .
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<html><body>
<div class="image" data-bbox="192 122 417 357"><img data-bbox="192 122 417 357"/><p class="caption" data-bbox="277 377 327 392">Figure 2: </p></div>
<p data-bbox="109 410 500 453">It is evident by construction that the integers $r$ and $s$ can be taken mod $d$ and mod $n$ respectively. Denote by $\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\in\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\geq0$ and $a+b+c>0$ . </p>
<p data-bbox="109 454 500 526">The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\mathit{m}}$ cycles $D_{1},\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$ does not disconnect the surface. Set $\mathcal{C}=\{C_{1},\ldots,C_{n}\}$ and $\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$ . </p>
<p data-bbox="109 526 500 597">If $m\,=\,n$ and if the cut along the cycles of $\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\scriptscriptstyle\mathcal{C}$ and $\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold. </p>
<p data-bbox="109 598 501 627">Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\boldsymbol{S}$ satisfying the following conditions: </p>
<p data-bbox="118 636 323 651">(1) the set $\mathcal{D}$ contains exactly $n$ cycles; </p>
<p data-bbox="118 660 486 675">(2) the surface $T_{n}^{'}$ is not disconnected by the cut along the cycles of $\mathcal{D}$ . </p>
</body></html>
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0003042v1
| 4 | 612 | 792 | 1,275 | 1,650 |
[{"type": "image", "img_path": "images/b6d2ab65c37ca0f1f4339baa89388725277b109f5a631747678d1b5fa2768fbe.jpg", "img_caption": ["Figure 2: "], "img_footnote": [], "page_idx": 4}, {"type": "text", "text": "It is evident by construction that the integers $r$ and $s$ can be taken mod $d$ and mod $n$ respectively. Denote by $\\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\\in\\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\\geq0$ and $a+b+c>0$ . ", "page_idx": 4}, {"type": "text", "text": "The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\\mathit{m}}$ cycles $D_{1},\\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\\prime}=C_{i-s}^{\\prime\\prime}$ does not disconnect the surface. Set $\\mathcal{C}=\\{C_{1},\\ldots,C_{n}\\}$ and $\\mathcal{D}=\\{D_{1},\\dotsc,D_{m}\\}$ . ", "page_idx": 4}, {"type": "text", "text": "If $m\\,=\\,n$ and if the cut along the cycles of $\\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\\scriptscriptstyle\\mathcal{C}$ and $\\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold. ", "page_idx": 4}, {"type": "text", "text": "Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\\boldsymbol{S}$ satisfying the following conditions: ", "page_idx": 4}, {"type": "text", "text": "(1) the set $\\mathcal{D}$ contains exactly $n$ cycles; ", "page_idx": 4}, {"type": "text", "text": "(2) the surface $T_{n}^{'}$ is not disconnected by the cut along the cycles of $\\mathcal{D}$ . ", "page_idx": 4}]
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The “open” Heegaard diagram $\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\sigma$ will be denoted by $H(\sigma)$ and $M(\sigma)$ respectively.
Remark 1. It is easy to see that not all the 6-tuples in $\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\geq1$ , give rise to exactly $a$ cycles in $\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ .
Consider now a 6-tuple $\sigma\,\in\,S$ . The graph $\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\Gamma^{\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\Omega=\cup_{i=1}^{n}C_{i}$ and $\Lambda=\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\Gamma$ . Since the gluing of the cycles of $\mathcal{C}^{\prime}$ and $\mathcal{C^{\prime\prime}}$ is invariant with respect to the rotation $\rho_{n}$ , the group $\mathcal{G}_{n}=<\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\Gamma^{\prime}$ is invariant under the rotation $\rho_{n}$ and $\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\mathcal{D}$ .
Lemma 1 a) Let $\sigma\;=\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\rho_{n}$ induces a cyclic permutation on the curves of $\mathcal{D}$ . Thus, if $D$ is a cycle of $\mathcal{D}$ , then ${\mathcal{D}}=\{\rho_{n}^{k-1}(D)|k=1,\ldots,n\}$ .
b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ .
Proof. a) First of all, note that $\rho_{n}(\Lambda)=\Lambda$ ; thus the group $\mathcal{G}_{n}$ also acts on the spaces $T_{n}\mathrm{~-~}\Lambda$ and $\Lambda$ (and hence on the set $\mathcal{D}$ ). If the 6-tuple $\sigma$ is admissible, then $T_{n}-\Lambda$ is connected, and hence the quotient $(T_{n}-\Lambda)/\mathcal{G}_{n}=$ $T_{n}/\mathcal{G}_{n}-\Lambda/\mathcal{G}_{n}$ must be connected too. This implies that $\Lambda/\mathcal{G}_{n}$ has a unique connected component. Since $\Lambda$ has exactly $n$ connected components, the cyclic group $\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\mathcal{D}$ .
b) Let $C,D\ \subset\ T_{1}$ the two curves $C\;=\;\Omega/\mathcal{G}_{n}$ and $D\,=\,\Lambda/\mathcal{G}_{n}$ . Then, the two systems of curves ${\mathcal{C}}=\{C\}$ and $\mathcal{D}=\{D\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\Gamma_{1}$ corresponding to $\sigma_{1}\,=\,(a,b,c,1,r,0)$ is the quotient of the graph $\Gamma_{n}$ corresponding to $\sigma=(a,b,c,n,r,s)$ , respect to $\mathcal{G}_{n}$ . Moreover, the gluings on $\Gamma_{n}$ are invariant respect to $\rho_{n}$ . Therefore, the gluings on $\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ .
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<p data-bbox="110 125 501 154">The “open” Heegaard diagram $\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\sigma$ will be denoted by $H(\sigma)$ and $M(\sigma)$ respectively. </p>
<p data-bbox="109 160 501 232">Remark 1. It is easy to see that not all the 6-tuples in $\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\geq1$ , give rise to exactly $a$ cycles in $\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ . </p>
<p data-bbox="109 238 500 383">Consider now a 6-tuple $\sigma\,\in\,S$ . The graph $\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\Gamma^{\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\Omega=\cup_{i=1}^{n}C_{i}$ and $\Lambda=\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\Gamma$ . Since the gluing of the cycles of $\mathcal{C}^{\prime}$ and $\mathcal{C^{\prime\prime}}$ is invariant with respect to the rotation $\rho_{n}$ , the group $\mathcal{G}_{n}=<\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\Gamma^{\prime}$ is invariant under the rotation $\rho_{n}$ and $\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\mathcal{D}$ . </p>
<p data-bbox="109 396 500 439">Lemma 1 a) Let $\sigma\;=\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\rho_{n}$ induces a cyclic permutation on the curves of $\mathcal{D}$ . Thus, if $D$ is a cycle of $\mathcal{D}$ , then ${\mathcal{D}}=\{\rho_{n}^{k-1}(D)|k=1,\ldots,n\}$ . </p>
<p data-bbox="110 440 501 483">b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ . </p>
<p data-bbox="109 496 501 596">Proof. a) First of all, note that $\rho_{n}(\Lambda)=\Lambda$ ; thus the group $\mathcal{G}_{n}$ also acts on the spaces $T_{n}\mathrm{~-~}\Lambda$ and $\Lambda$ (and hence on the set $\mathcal{D}$ ). If the 6-tuple $\sigma$ is admissible, then $T_{n}-\Lambda$ is connected, and hence the quotient $(T_{n}-\Lambda)/\mathcal{G}_{n}=$ $T_{n}/\mathcal{G}_{n}-\Lambda/\mathcal{G}_{n}$ must be connected too. This implies that $\Lambda/\mathcal{G}_{n}$ has a unique connected component. Since $\Lambda$ has exactly $n$ connected components, the cyclic group $\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\mathcal{D}$ . </p>
<p data-bbox="109 597 500 669">b) Let $C,D\ \subset\ T_{1}$ the two curves $C\;=\;\Omega/\mathcal{G}_{n}$ and $D\,=\,\Lambda/\mathcal{G}_{n}$ . Then, the two systems of curves ${\mathcal{C}}=\{C\}$ and $\mathcal{D}=\{D\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\Gamma_{1}$ corresponding to $\sigma_{1}\,=\,(a,b,c,1,r,0)$ is the quotient of the graph $\Gamma_{n}$ corresponding to $\sigma=(a,b,c,n,r,s)$ , respect to $\mathcal{G}_{n}$ . Moreover, the gluings on $\Gamma_{n}$ are invariant respect to $\rho_{n}$ . Therefore, the gluings on $\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ . </p>
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0003042v1
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[{"type": "text", "text": "The “open” Heegaard diagram $\\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\\sigma$ will be denoted by $H(\\sigma)$ and $M(\\sigma)$ respectively. ", "page_idx": 5}, {"type": "text", "text": "Remark 1. It is easy to see that not all the 6-tuples in $\\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\\geq1$ , give rise to exactly $a$ cycles in $\\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ . ", "page_idx": 5}, {"type": "text", "text": "Consider now a 6-tuple $\\sigma\\,\\in\\,S$ . The graph $\\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\\Gamma^{\\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\\Omega=\\cup_{i=1}^{n}C_{i}$ and $\\Lambda=\\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\\Gamma$ . Since the gluing of the cycles of $\\mathcal{C}^{\\prime}$ and $\\mathcal{C^{\\prime\\prime}}$ is invariant with respect to the rotation $\\rho_{n}$ , the group $\\mathcal{G}_{n}=<\\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\\Gamma^{\\prime}$ is invariant under the rotation $\\rho_{n}$ and $\\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "Lemma 1 a) Let $\\sigma\\;=\\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\\rho_{n}$ induces a cyclic permutation on the curves of $\\mathcal{D}$ . Thus, if $D$ is a cycle of $\\mathcal{D}$ , then ${\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}$ . ", "page_idx": 5}, {"type": "text", "text": "b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. a) First of all, note that $\\rho_{n}(\\Lambda)=\\Lambda$ ; thus the group $\\mathcal{G}_{n}$ also acts on the spaces $T_{n}\\mathrm{~-~}\\Lambda$ and $\\Lambda$ (and hence on the set $\\mathcal{D}$ ). If the 6-tuple $\\sigma$ is admissible, then $T_{n}-\\Lambda$ is connected, and hence the quotient $(T_{n}-\\Lambda)/\\mathcal{G}_{n}=$ $T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}$ must be connected too. This implies that $\\Lambda/\\mathcal{G}_{n}$ has a unique connected component. Since $\\Lambda$ has exactly $n$ connected components, the cyclic group $\\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "b) Let $C,D\\ \\subset\\ T_{1}$ the two curves $C\\;=\\;\\Omega/\\mathcal{G}_{n}$ and $D\\,=\\,\\Lambda/\\mathcal{G}_{n}$ . Then, the two systems of curves ${\\mathcal{C}}=\\{C\\}$ and $\\mathcal{D}=\\{D\\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\\Gamma_{1}$ corresponding to $\\sigma_{1}\\,=\\,(a,b,c,1,r,0)$ is the quotient of the graph $\\Gamma_{n}$ corresponding to $\\sigma=(a,b,c,n,r,s)$ , respect to $\\mathcal{G}_{n}$ . Moreover, the gluings on $\\Gamma_{n}$ are invariant respect to $\\rho_{n}$ . Therefore, the gluings on $\\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The “open” Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}], "index": 32}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The “open” Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [109, 128, 500, 156]}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4, "bbox_fs": [109, 163, 500, 234]}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5, "bbox_fs": [108, 239, 501, 384]}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [108, 395, 499, 441]}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [111, 441, 501, 486]}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26, "bbox_fs": [109, 497, 501, 599]}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 142], "spans": [{"bbox": [110, 128, 187, 142], "score": 1.0, "content": "the gluings on ", "type": "text", "cross_page": true}, {"bbox": [187, 130, 199, 140], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12, "cross_page": true}, {"bbox": [199, 128, 500, 142], "score": 1.0, "content": " give rise to the Heegaard diagram above. This show that", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 501, 156], "spans": [{"bbox": [110, 142, 170, 156], "score": 1.0, "content": "the 6-tuple ", "type": "text", "cross_page": true}, {"bbox": [170, 147, 181, 154], "score": 0.88, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 11, "cross_page": true}, {"bbox": [182, 142, 328, 156], "score": 1.0, "content": " is admissible and obviously ", "type": "text", "cross_page": true}, {"bbox": [329, 143, 407, 155], "score": 0.94, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 78, "cross_page": true}, {"bbox": [408, 142, 501, 156], "score": 1.0, "content": " is the quotient of", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 155, 281, 172], "spans": [{"bbox": [110, 158, 190, 170], "score": 0.93, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 80, "cross_page": true}, {"bbox": [190, 155, 248, 172], "score": 1.0, "content": " respect to ", "type": "text", "cross_page": true}, {"bbox": [248, 158, 261, 169], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [261, 155, 281, 172], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 2}], "index": 32, "bbox_fs": [109, 599, 500, 671]}]}
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[{"type": "text", "bbox": [110, 125, 501, 154], "content": "The “open” Heegaard diagram and the Dunwoody manifold associated to the admissible 6-tuple will be denoted by and respectively.", "index": 0}, {"type": "text", "bbox": [109, 160, 501, 232], "content": "Remark 1. It is easy to see that not all the 6-tuples in are admissible. For example, the 6-tuples , with , give rise to exactly cycles in ; thus, they are not admissible if . The 6-tuples are not admissible if is even, since, in this case, we obtain exactly one cycle , but the cut along it disconnects the torus .", "index": 1}, {"type": "text", "bbox": [109, 238, 500, 383], "content": "Consider now a 6-tuple . The graph becomes, via the gluing quotient map, a regular 4-valent graph denoted by embedded in . Its vertices are the intersection points of the spaces and ; hence they inherit the labelling of the corresponding glued vertices of . Since the gluing of the cycles of and is invariant with respect to the rotation , the group naturally induces a cyclic action of order on such that the quotient is homeomorphic to a torus. The labelling of the vertices of is invariant under the rotation and (mod ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of .", "index": 2}, {"type": "text", "bbox": [109, 396, 500, 439], "content": "Lemma 1 a) Let be an admissible 6-tuple. Then induces a cyclic permutation on the curves of . Thus, if is a cycle of , then .", "index": 3}, {"type": "text", "bbox": [110, 440, 501, 483], "content": "b) If is admissible, then also is admissible and the Heegaard diagram is the quotient of the Heegaard diagram respect to .", "index": 4}, {"type": "text", "bbox": [109, 496, 501, 596], "content": "Proof. a) First of all, note that ; thus the group also acts on the spaces and (and hence on the set ). If the 6-tuple is admissible, then is connected, and hence the quotient must be connected too. This implies that has a unique connected component. Since has exactly connected components, the cyclic group of order defines a simply transitive cyclic action on the cycles of .", "index": 5}, {"type": "text", "bbox": [109, 597, 500, 669], "content": "b) Let the two curves and . Then, the two systems of curves and on define a Heegaard diagram of genus one. The graph corresponding to is the quotient of the graph corresponding to , respect to . Moreover, the gluings on are invariant respect to . Therefore, the gluings on give rise to the Heegaard diagram above. This show that the 6-tuple is admissible and obviously is the quotient of respect to .", "index": 6}]
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[{"bbox": [127, 128, 500, 142], "content": "The “open” Heegaard diagram and the Dunwoody manifold associated", "parent_index": 0, "line_index": 0}, {"bbox": [109, 142, 498, 156], "content": "to the admissible 6-tuple will be denoted by and respectively.", "parent_index": 0, "line_index": 1}, {"bbox": [110, 163, 500, 176], "content": "Remark 1. It is easy to see that not all the 6-tuples in are admissible. For", "parent_index": 1, "line_index": 0}, {"bbox": [110, 178, 500, 191], "content": "example, the 6-tuples , with , give rise to exactly cycles", "parent_index": 1, "line_index": 1}, {"bbox": [109, 191, 500, 206], "content": "in ; thus, they are not admissible if . The 6-tuples are", "parent_index": 1, "line_index": 2}, {"bbox": [110, 206, 499, 219], "content": "not admissible if is even, since, in this case, we obtain exactly one cycle", "parent_index": 1, "line_index": 3}, {"bbox": [110, 220, 365, 234], "content": ", but the cut along it disconnects the torus .", "parent_index": 1, "line_index": 4}, {"bbox": [127, 239, 500, 256], "content": "Consider now a 6-tuple . The graph becomes, via the gluing", "parent_index": 2, "line_index": 0}, {"bbox": [110, 256, 501, 269], "content": "quotient map, a regular 4-valent graph denoted by embedded in . Its", "parent_index": 2, "line_index": 1}, {"bbox": [108, 267, 501, 285], "content": "vertices are the intersection points of the spaces and ;", "parent_index": 2, "line_index": 2}, {"bbox": [109, 284, 501, 298], "content": "hence they inherit the labelling of the corresponding glued vertices of . Since", "parent_index": 2, "line_index": 3}, {"bbox": [110, 299, 500, 312], "content": "the gluing of the cycles of and is invariant with respect to the rotation", "parent_index": 2, "line_index": 4}, {"bbox": [110, 313, 499, 328], "content": ", the group naturally induces a cyclic action of order on", "parent_index": 2, "line_index": 5}, {"bbox": [110, 328, 500, 342], "content": "such that the quotient is homeomorphic to a torus. The labelling", "parent_index": 2, "line_index": 6}, {"bbox": [109, 341, 501, 357], "content": "of the vertices of is invariant under the rotation and (mod", "parent_index": 2, "line_index": 7}, {"bbox": [110, 357, 499, 371], "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "parent_index": 2, "line_index": 8}, {"bbox": [110, 371, 263, 384], "content": "also holds for the cycles of .", "parent_index": 2, "line_index": 9}, {"bbox": [108, 395, 498, 414], "content": "Lemma 1 a) Let be an admissible 6-tuple. Then", "parent_index": 3, "line_index": 0}, {"bbox": [111, 412, 499, 426], "content": "induces a cyclic permutation on the curves of . Thus, if is a cycle of ,", "parent_index": 3, "line_index": 1}, {"bbox": [110, 426, 285, 441], "content": "then .", "parent_index": 3, "line_index": 2}, {"bbox": [127, 441, 501, 455], "content": "b) If is admissible, then also is admissible", "parent_index": 4, "line_index": 0}, {"bbox": [111, 456, 501, 470], "content": "and the Heegaard diagram is the quotient of the Heegaard", "parent_index": 4, "line_index": 1}, {"bbox": [111, 469, 309, 486], "content": "diagram respect to .", "parent_index": 4, "line_index": 2}, {"bbox": [126, 497, 501, 512], "content": "Proof. a) First of all, note that ; thus the group also acts", "parent_index": 5, "line_index": 0}, {"bbox": [110, 513, 500, 526], "content": "on the spaces and (and hence on the set ). If the 6-tuple is", "parent_index": 5, "line_index": 1}, {"bbox": [109, 526, 501, 541], "content": "admissible, then is connected, and hence the quotient", "parent_index": 5, "line_index": 2}, {"bbox": [110, 542, 499, 555], "content": "must be connected too. This implies that has a unique", "parent_index": 5, "line_index": 3}, {"bbox": [110, 556, 500, 569], "content": "connected component. Since has exactly connected components, the", "parent_index": 5, "line_index": 4}, {"bbox": [110, 570, 500, 583], "content": "cyclic group of order defines a simply transitive cyclic action on the", "parent_index": 5, "line_index": 5}, {"bbox": [110, 585, 172, 599], "content": "cycles of .", "parent_index": 5, "line_index": 6}, {"bbox": [127, 599, 499, 613], "content": "b) Let the two curves and . Then,", "parent_index": 6, "line_index": 0}, {"bbox": [110, 613, 500, 627], "content": "the two systems of curves and on define a Heegaard", "parent_index": 6, "line_index": 1}, {"bbox": [110, 627, 499, 642], "content": "diagram of genus one. The graph corresponding to", "parent_index": 6, "line_index": 2}, {"bbox": [109, 642, 500, 657], "content": "is the quotient of the graph corresponding to , respect", "parent_index": 6, "line_index": 3}, {"bbox": [109, 657, 500, 671], "content": "to . Moreover, the gluings on are invariant respect to . Therefore,", "parent_index": 6, "line_index": 4}, {"bbox": [110, 128, 500, 142], "content": "the gluings on give rise to the Heegaard diagram above. This show that", "parent_index": 6, "line_index": 5}, {"bbox": [110, 142, 501, 156], "content": "the 6-tuple is admissible and obviously is the quotient of", "parent_index": 6, "line_index": 6}, {"bbox": [110, 155, 281, 172], "content": "respect to .", "parent_index": 6, "line_index": 7}]
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[{"bbox": [288, 129, 295, 138], "content": "\\Gamma", "parent_index": 0, "subtype": "inline"}, {"bbox": [241, 147, 248, 153], "content": "\\sigma", "parent_index": 0, "subtype": "inline"}, {"bbox": [351, 143, 378, 156], "content": "H(\\sigma)", "parent_index": 0, "subtype": "inline"}, {"bbox": [404, 143, 433, 156], "content": "M(\\sigma)", "parent_index": 0, "subtype": "inline"}, {"bbox": [391, 164, 399, 173], "content": "\\boldsymbol{S}", "parent_index": 1, "subtype": "inline"}, {"bbox": [223, 178, 294, 190], "content": "(a,0,a,1,a,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [326, 179, 355, 189], "content": "a\\geq1", "parent_index": 1, "subtype": "inline"}, {"bbox": [460, 182, 466, 187], "content": "a", "parent_index": 1, "subtype": "inline"}, {"bbox": [124, 194, 134, 202], "content": "\\mathcal{D}", "parent_index": 1, "subtype": "inline"}, {"bbox": [304, 194, 333, 202], "content": "a>1", "parent_index": 1, "subtype": "inline"}, {"bbox": [410, 192, 479, 205], "content": "(1,0,c,1,2,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [201, 211, 207, 216], "content": "c", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 222, 125, 232], "content": "D_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [348, 222, 360, 232], "content": "T_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [255, 243, 290, 252], "content": "\\sigma\\,\\in\\,S", "parent_index": 2, "subtype": "inline"}, {"bbox": [362, 243, 369, 251], "content": "\\Gamma", "parent_index": 2, "subtype": "inline"}, {"bbox": [378, 257, 388, 266], "content": "\\Gamma^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [462, 257, 475, 267], "content": "T_{n}^{'}", "parent_index": 2, "subtype": "inline"}, {"bbox": [352, 271, 410, 283], "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [434, 271, 496, 285], "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "parent_index": 2, "subtype": "inline"}, {"bbox": [457, 286, 464, 294], "content": "\\Gamma", "parent_index": 2, "subtype": "inline"}, {"bbox": [245, 300, 255, 309], "content": "\\mathcal{C}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [281, 300, 293, 309], "content": "\\mathcal{C^{\\prime\\prime}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 318, 122, 326], "content": "\\rho_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [182, 315, 244, 326], "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "parent_index": 2, "subtype": "inline"}, {"bbox": [459, 318, 466, 323], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [486, 315, 499, 325], "content": "T_{n}^{'}", "parent_index": 2, "subtype": "inline"}, {"bbox": [225, 329, 284, 341], "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [195, 343, 205, 352], "content": "\\Gamma^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [362, 347, 374, 355], "content": "\\rho_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [398, 343, 469, 355], "content": "\\rho_{n}(C_{i})=C_{i+1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 361, 118, 367], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [249, 372, 259, 381], "content": "\\mathcal{D}", "parent_index": 2, "subtype": "inline"}, {"bbox": [209, 399, 308, 411], "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "parent_index": 3, "subtype": "inline"}, {"bbox": [487, 403, 498, 411], "content": "\\rho_{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [344, 414, 354, 423], "content": "\\mathcal{D}", "parent_index": 3, "subtype": "inline"}, {"bbox": [407, 414, 417, 423], "content": "D", "parent_index": 3, "subtype": "inline"}, {"bbox": [486, 415, 496, 423], "content": "\\mathcal{D}", "parent_index": 3, "subtype": "inline"}, {"bbox": [136, 427, 281, 441], "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [157, 442, 226, 455], "content": "(a,b,c,n,r,s)", "parent_index": 4, "subtype": "inline"}, {"bbox": [359, 442, 428, 455], "content": "(a,b,c,1,r,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [252, 457, 331, 469], "content": "H(a,b,c,1,r,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [155, 471, 235, 484], "content": "H(a,b,c,n,r,s)", "parent_index": 4, "subtype": "inline"}, {"bbox": [291, 472, 304, 483], "content": "\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [298, 499, 352, 511], "content": "\\rho_{n}(\\Lambda)=\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [438, 500, 451, 510], "content": "\\mathcal{G}_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [185, 514, 222, 524], "content": "T_{n}\\mathrm{~-~}\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [249, 514, 258, 523], "content": "\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [380, 514, 390, 523], "content": "\\mathcal{D}", "parent_index": 5, "subtype": "inline"}, {"bbox": [479, 517, 486, 523], "content": "\\sigma", "parent_index": 5, "subtype": "inline"}, {"bbox": [196, 528, 231, 539], "content": "T_{n}-\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [424, 528, 501, 540], "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 542, 183, 555], "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [403, 542, 430, 555], "content": "\\Lambda/\\mathcal{G}_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [267, 557, 275, 566], "content": "\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [344, 560, 352, 566], "content": "n", "parent_index": 5, "subtype": "inline"}, {"bbox": [178, 572, 190, 582], "content": "\\mathcal{G}_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [241, 575, 249, 580], "content": "n", "parent_index": 5, "subtype": "inline"}, {"bbox": [157, 587, 167, 595], "content": "\\mathcal{D}", "parent_index": 5, "subtype": "inline"}, {"bbox": [167, 601, 223, 612], "content": "C,D\\ \\subset\\ T_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [311, 600, 369, 613], "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [398, 600, 456, 613], "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [247, 614, 293, 627], "content": "{\\mathcal{C}}=\\{C\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [320, 614, 370, 627], "content": "\\mathcal{D}=\\{D\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [390, 615, 402, 626], "content": "T_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [292, 630, 304, 640], "content": "\\Gamma_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [400, 629, 499, 641], "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "parent_index": 6, "subtype": "inline"}, {"bbox": [256, 644, 269, 654], "content": "\\Gamma_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [363, 643, 455, 655], "content": "\\sigma=(a,b,c,n,r,s)", "parent_index": 6, "subtype": "inline"}, {"bbox": [125, 658, 138, 669], "content": "\\mathcal{G}_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [281, 658, 294, 669], "content": "\\Gamma_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [424, 662, 436, 669], "content": "\\rho_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [187, 130, 199, 140], "content": "\\Gamma_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [170, 147, 181, 154], "content": "\\sigma_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [329, 143, 407, 155], "content": "H(a,b,c,1,r,0)", "parent_index": 6, "subtype": "inline"}, {"bbox": [110, 158, 190, 170], "content": "H(a,b,c,n,r,s)", "parent_index": 6, "subtype": "inline"}, {"bbox": [248, 158, 261, 169], "content": "\\mathcal{G}_{n}", "parent_index": 6, "subtype": "inline"}]
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Remark 2. More generally, given two positive integer $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\cal H}(a,b,c,r,n^{\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\prime}$ .
It is easy to see that, for admissible 6-tuples, each cycle in $\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\Gamma$ (in fact, $\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs).
An important consequence of point a) of Lemma $1$ is that, if $\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\sigma)$ induced by the Heegaard diagram $H(\sigma)$ is cyclic.
To see this, let $\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\mathcal{D}$ containing $v$ and by $v^{\prime}$ the vertex of $C_{1}^{\prime}$ corresponding to $v$ . Orient the arc $e^{\prime}\in A$ of the graph $\Gamma$ containing $v^{\prime}$ so that $v^{\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\mathcal{D}$ induce an orientation on the arcs of the graph $\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\prime}$ and $C^{\prime\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ , the graph $\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\Gamma$ .
Let now $w\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\prime},e_{2},\ldots,e_{d}$ of $\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\prime\prime}$ . Note that the word $\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\leq k\leq n$ (roughly speaking, the automorphism $\theta_{n}$ is “geometrically” realized by $\rho_{n}$ ).
This proves that each admissible 6-tuple $\sigma$ uniquely defines, via the associated Heegaard diagram $H(\sigma)$ , a word $w=w(\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\sigma)$ . Note that the sequence of the exponents in the word $w(\sigma)$ , and hence its exponentsum $\varepsilon_{w(\sigma)}$ , only depends on the integers $a,b,c,r$ .
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<p data-bbox="110 174 501 232">Remark 2. More generally, given two positive integer $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\cal H}(a,b,c,r,n^{\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\prime}$ . </p>
<p data-bbox="110 239 500 281">It is easy to see that, for admissible 6-tuples, each cycle in $\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\Gamma$ (in fact, $\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs). </p>
<p data-bbox="110 282 500 325">An important consequence of point a) of Lemma $1$ is that, if $\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\sigma)$ induced by the Heegaard diagram $H(\sigma)$ is cyclic. </p>
<p data-bbox="109 326 500 484">To see this, let $\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\mathcal{D}$ containing $v$ and by $v^{\prime}$ the vertex of $C_{1}^{\prime}$ corresponding to $v$ . Orient the arc $e^{\prime}\in A$ of the graph $\Gamma$ containing $v^{\prime}$ so that $v^{\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\mathcal{D}$ induce an orientation on the arcs of the graph $\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\prime}$ and $C^{\prime\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ , the graph $\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\Gamma$ . </p>
<p data-bbox="109 485 501 599">Let now $w\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\prime},e_{2},\ldots,e_{d}$ of $\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\prime\prime}$ . Note that the word $\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\leq k\leq n$ (roughly speaking, the automorphism $\theta_{n}$ is “geometrically” realized by $\rho_{n}$ ). </p>
<p data-bbox="109 601 500 672">This proves that each admissible 6-tuple $\sigma$ uniquely defines, via the associated Heegaard diagram $H(\sigma)$ , a word $w=w(\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\sigma)$ . Note that the sequence of the exponents in the word $w(\sigma)$ , and hence its exponentsum $\varepsilon_{w(\sigma)}$ , only depends on the integers $a,b,c,r$ . </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 6}, {"type": "text", "text": "Remark 2. More generally, given two positive integer $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\\cal H}(a,b,c,r,n^{\\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\\prime}$ . ", "page_idx": 6}, {"type": "text", "text": "It is easy to see that, for admissible 6-tuples, each cycle in $\\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\\Gamma$ (in fact, $\\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs). ", "page_idx": 6}, {"type": "text", "text": "An important consequence of point a) of Lemma $1$ is that, if $\\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\\sigma)$ induced by the Heegaard diagram $H(\\sigma)$ is cyclic. ", "page_idx": 6}, {"type": "text", "text": "To see this, let $\\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\\mathcal{D}$ containing $v$ and by $v^{\\prime}$ the vertex of $C_{1}^{\\prime}$ corresponding to $v$ . Orient the arc $e^{\\prime}\\in A$ of the graph $\\Gamma$ containing $v^{\\prime}$ so that $v^{\\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\\mathcal{D}$ induce an orientation on the arcs of the graph $\\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\\prime}$ and $C^{\\prime\\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ , the graph $\\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\\Gamma$ . ", "page_idx": 6}, {"type": "text", "text": "Let now $w\\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\\prime},e_{2},\\ldots,e_{d}$ of $\\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\\prime\\prime}$ . Note that the word $\\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\\leq k\\leq n$ (roughly speaking, the automorphism $\\theta_{n}$ is “geometrically” realized by $\\rho_{n}$ ). ", "page_idx": 6}, {"type": "text", "text": "This proves that each admissible 6-tuple $\\sigma$ uniquely defines, via the associated Heegaard diagram $H(\\sigma)$ , a word $w=w(\\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\\sigma)$ . Note that the sequence of the exponents in the word $w(\\sigma)$ , and hence its exponentsum $\\varepsilon_{w(\\sigma)}$ , only depends on the integers $a,b,c,r$ . ", "page_idx": 6}]
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This show that", "type": "text"}], "index": 0}, {"bbox": [110, 142, 501, 156], "spans": [{"bbox": [110, 142, 170, 156], "score": 1.0, "content": "the 6-tuple ", "type": "text"}, {"bbox": [170, 147, 181, 154], "score": 0.88, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [182, 142, 328, 156], "score": 1.0, "content": " is admissible and obviously ", "type": "text"}, {"bbox": [329, 143, 407, 155], "score": 0.94, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [408, 142, 501, 156], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 1}, {"bbox": [110, 155, 281, 172], "spans": [{"bbox": [110, 158, 190, 170], "score": 0.93, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [190, 155, 248, 172], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [248, 158, 261, 169], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [261, 155, 281, 172], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [110, 174, 501, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 401, 191], "score": 1.0, "content": "Remark 2. More generally, given two positive integer ", "type": "text"}, {"bbox": [401, 182, 408, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [408, 176, 435, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [436, 178, 446, 187], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 176, 500, 191], "score": 1.0, "content": " such that", "type": "text"}], "index": 3}, {"bbox": [110, 191, 500, 205], "spans": [{"bbox": [110, 193, 120, 202], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [120, 191, 163, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [163, 196, 171, 202], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [171, 191, 188, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [188, 192, 257, 205], "score": 0.94, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [257, 191, 359, 205], "score": 1.0, "content": " is admissible, then ", "type": "text"}, {"bbox": [359, 192, 430, 205], "score": 0.95, "content": "(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [430, 191, 500, 205], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 206, 502, 220], "spans": [{"bbox": [109, 206, 318, 220], "score": 1.0, "content": "too. Moreover, the Heegaard diagram ", "type": "text"}, {"bbox": [319, 207, 401, 219], "score": 0.93, "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [401, 206, 502, 220], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 5}, {"bbox": [110, 220, 462, 235], "spans": [{"bbox": [110, 221, 189, 234], "score": 0.93, "content": "H(a,b,c,r,n,b)", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [190, 220, 435, 235], "score": 1.0, "content": " respect to the action of a cyclic group of order ", "type": "text"}, {"bbox": [435, 221, 458, 234], "score": 0.94, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [458, 220, 462, 235], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5}, {"type": "text", "bbox": [110, 239, 500, 281], "lines": [{"bbox": [127, 241, 499, 254], "spans": [{"bbox": [127, 241, 433, 254], "score": 1.0, "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "type": "text"}, {"bbox": [433, 243, 443, 251], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [443, 241, 492, 254], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [493, 243, 499, 251], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}], "index": 7}, {"bbox": [110, 255, 500, 269], "spans": [{"bbox": [110, 255, 400, 269], "score": 1.0, "content": "vertices with different labels and is composed by exactly ", "type": "text"}, {"bbox": [401, 257, 407, 266], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [407, 255, 447, 269], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [447, 257, 455, 266], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [455, 255, 500, 269], "score": 1.0, "content": " (in fact,", "type": "text"}], "index": 8}, {"bbox": [110, 270, 384, 284], "spans": [{"bbox": [110, 272, 123, 280], "score": 0.85, "content": "\\mathit{2a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [123, 270, 208, 284], "score": 1.0, "content": " horizontal arcs, ", "type": "text"}, {"bbox": [208, 271, 213, 280], "score": 0.89, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [214, 270, 304, 284], "score": 1.0, "content": " oblique arcs and ", "type": "text"}, {"bbox": [305, 275, 310, 280], "score": 0.89, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [311, 270, 384, 284], "score": 1.0, "content": " vertical arcs).", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [110, 282, 500, 325], "lines": [{"bbox": [127, 285, 499, 297], "spans": [{"bbox": [127, 285, 382, 297], "score": 1.0, "content": "An important consequence of point a) of Lemma ", "type": "text"}, {"bbox": [382, 286, 388, 294], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [388, 285, 443, 297], "score": 1.0, "content": " is that, if ", "type": "text"}, {"bbox": [443, 289, 450, 294], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [451, 285, 499, 297], "score": 1.0, "content": " is an ad-", "type": "text"}], "index": 10}, {"bbox": [110, 298, 500, 312], "spans": [{"bbox": [110, 298, 426, 312], "score": 1.0, "content": "missible 6-tuple, the presentation of the fundamental group of ", "type": "text"}, {"bbox": [427, 299, 456, 312], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [456, 298, 500, 312], "score": 1.0, "content": " induced", "type": "text"}], "index": 11}, {"bbox": [110, 313, 317, 326], "spans": [{"bbox": [110, 313, 243, 326], "score": 1.0, "content": "by the Heegaard diagram ", "type": "text"}, {"bbox": [244, 314, 271, 326], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [271, 313, 317, 326], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [109, 326, 500, 484], "lines": [{"bbox": [127, 327, 499, 341], "spans": [{"bbox": [127, 327, 206, 341], "score": 1.0, "content": "To see this, let ", "type": "text"}, {"bbox": [206, 333, 213, 338], "score": 0.88, "content": "\\upsilon", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 327, 403, 341], "score": 1.0, "content": " be the vertex belonging to the cycle ", "type": "text"}, {"bbox": [403, 329, 417, 340], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [417, 327, 499, 341], "score": 1.0, "content": " and labelled by", "type": "text"}], "index": 13}, {"bbox": [110, 342, 500, 356], "spans": [{"bbox": [110, 344, 159, 353], "score": 0.93, "content": "a+b+1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [159, 342, 222, 356], "score": 1.0, "content": "; denote by ", "type": "text"}, {"bbox": [222, 344, 237, 354], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [237, 342, 308, 356], "score": 1.0, "content": " the curve of ", "type": "text"}, {"bbox": [309, 344, 318, 352], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [319, 342, 380, 356], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [381, 347, 387, 352], "score": 0.89, "content": "v", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [387, 342, 432, 356], "score": 1.0, "content": " and by ", "type": "text"}, {"bbox": [433, 343, 442, 352], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [442, 342, 500, 356], "score": 1.0, "content": " the vertex", "type": "text"}], "index": 14}, {"bbox": [110, 356, 500, 370], "spans": [{"bbox": [110, 356, 124, 370], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [124, 358, 137, 370], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [137, 356, 232, 370], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [232, 361, 239, 367], "score": 0.89, "content": "v", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [239, 356, 327, 370], "score": 1.0, "content": ". Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". The letters of ", "type": "text"}, {"bbox": [232, 520, 241, 525], "score": 0.88, "content": "w", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [241, 515, 500, 529], "score": 1.0, "content": " are in one-to-one correspondence with the oriented", "type": "text"}], "index": 26}, {"bbox": [110, 530, 500, 543], "spans": [{"bbox": [110, 530, 134, 543], "score": 1.0, "content": "arcs ", "type": "text"}, {"bbox": [135, 534, 146, 542], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [146, 530, 298, 543], "score": 1.0, "content": "; more precisely, the letter of ", "type": "text"}, {"bbox": [298, 534, 307, 540], "score": 0.89, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [307, 530, 401, 543], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [401, 534, 412, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [412, 530, 428, 543], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [428, 534, 438, 542], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [438, 530, 453, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [453, 534, 464, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [464, 530, 500, 543], "score": 1.0, "content": " comes", "type": "text"}], "index": 27}, {"bbox": [109, 543, 501, 560], "spans": [{"bbox": [109, 543, 209, 560], "score": 1.0, "content": "out from the cycle ", "type": "text"}, {"bbox": [209, 545, 221, 558], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [222, 543, 260, 560], "score": 1.0, "content": " and is ", "type": "text"}, {"bbox": [261, 544, 279, 558], "score": 0.95, "content": "x_{i}^{-1}", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [279, 543, 293, 560], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [294, 549, 304, 556], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [305, 543, 442, 560], "score": 1.0, "content": " comes out from the cycle ", "type": "text"}, {"bbox": [443, 545, 465, 558], "score": 0.93, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [465, 543, 501, 560], "score": 1.0, "content": ". Note", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 573], "spans": [{"bbox": [109, 558, 181, 573], "score": 1.0, "content": "that the word ", "type": "text"}, {"bbox": [181, 559, 221, 572], "score": 0.94, "content": "\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [221, 558, 352, 573], "score": 1.0, "content": " in the cyclic presentation ", "type": "text"}, {"bbox": [352, 560, 385, 572], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [386, 558, 500, 573], "score": 1.0, "content": " is obtained by reading", "type": "text"}], "index": 29}, {"bbox": [109, 572, 499, 588], "spans": [{"bbox": [109, 572, 160, 588], "score": 1.0, "content": "the cycle ", "type": "text"}, {"bbox": [160, 575, 175, 586], "score": 0.92, "content": "D_{k}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [176, 572, 344, 588], "score": 1.0, "content": " along the given orientation, for ", "type": "text"}, {"bbox": [345, 575, 399, 585], "score": 0.92, "content": "1\\leq k\\leq n", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [400, 572, 499, 588], "score": 1.0, "content": " (roughly speaking,", "type": "text"}], "index": 30}, {"bbox": [109, 587, 398, 602], "spans": [{"bbox": [109, 587, 207, 602], "score": 1.0, "content": "the automorphism ", "type": "text"}, {"bbox": [208, 589, 219, 600], "score": 0.92, "content": "\\theta_{n}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [219, 587, 377, 602], "score": 1.0, "content": " is “geometrically” realized by ", "type": "text"}, {"bbox": [377, 592, 389, 600], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [389, 587, 398, 602], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 27.5}, {"type": "text", "bbox": [109, 601, 500, 672], "lines": [{"bbox": [127, 601, 499, 616], "spans": [{"bbox": [127, 601, 336, 616], "score": 1.0, "content": "This proves that each admissible 6-tuple ", "type": "text"}, {"bbox": [336, 607, 343, 612], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [344, 601, 499, 616], "score": 1.0, "content": " uniquely defines, via the asso-", "type": "text"}], "index": 32}, {"bbox": [110, 617, 499, 630], "spans": [{"bbox": [110, 617, 242, 630], "score": 1.0, "content": "ciated Heegaard diagram ", "type": "text"}, {"bbox": [243, 617, 270, 630], "score": 0.94, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [270, 617, 316, 630], "score": 1.0, "content": ", a word ", "type": "text"}, {"bbox": [316, 617, 366, 630], "score": 0.95, "content": "w=w(\\sigma)", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [367, 617, 499, 630], "score": 1.0, "content": " and a cyclic presentation", "type": "text"}], "index": 33}, {"bbox": [110, 631, 500, 644], "spans": [{"bbox": [110, 632, 143, 644], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [144, 631, 434, 644], "score": 1.0, "content": " for the fundamental group of the Dunwoody manifold ", "type": "text"}, {"bbox": [434, 632, 463, 644], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [464, 631, 500, 644], "score": 1.0, "content": ". Note", "type": "text"}], "index": 34}, {"bbox": [110, 645, 499, 659], "spans": [{"bbox": [110, 645, 348, 659], "score": 1.0, "content": "that the sequence of the exponents in the word ", "type": "text"}, {"bbox": [348, 646, 373, 658], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [373, 645, 499, 659], "score": 1.0, "content": ", and hence its exponent-", "type": "text"}], "index": 35}, {"bbox": [109, 659, 359, 675], "spans": [{"bbox": [109, 659, 134, 675], "score": 1.0, "content": "sum ", "type": "text"}, {"bbox": [135, 664, 159, 674], "score": 0.92, "content": "\\varepsilon_{w(\\sigma)}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [160, 659, 316, 675], "score": 1.0, "content": ", only depends on the integers ", "type": "text"}, {"bbox": [317, 661, 354, 672], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [355, 659, 359, 675], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [300, 692, 310, 705], "spans": [{"bbox": [300, 692, 310, 705], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 169], "lines": [], "index": 1, "bbox_fs": [110, 128, 501, 172], "lines_deleted": true}, {"type": "text", "bbox": [110, 174, 501, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 401, 191], "score": 1.0, "content": "Remark 2. More generally, given two positive integer ", "type": "text"}, {"bbox": [401, 182, 408, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [408, 176, 435, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [436, 178, 446, 187], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 176, 500, 191], "score": 1.0, "content": " such that", "type": "text"}], "index": 3}, {"bbox": [110, 191, 500, 205], "spans": [{"bbox": [110, 193, 120, 202], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [120, 191, 163, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [163, 196, 171, 202], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [171, 191, 188, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [188, 192, 257, 205], "score": 0.94, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [257, 191, 359, 205], "score": 1.0, "content": " is admissible, then ", "type": "text"}, {"bbox": [359, 192, 430, 205], "score": 0.95, "content": "(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [430, 191, 500, 205], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 206, 502, 220], "spans": [{"bbox": [109, 206, 318, 220], "score": 1.0, "content": "too. Moreover, the Heegaard diagram ", "type": "text"}, {"bbox": [319, 207, 401, 219], "score": 0.93, "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [401, 206, 502, 220], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 5}, {"bbox": [110, 220, 462, 235], "spans": [{"bbox": [110, 221, 189, 234], "score": 0.93, "content": "H(a,b,c,r,n,b)", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [190, 220, 435, 235], "score": 1.0, "content": " respect to the action of a cyclic group of order ", "type": "text"}, {"bbox": [435, 221, 458, 234], "score": 0.94, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [458, 220, 462, 235], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "bbox_fs": [109, 176, 502, 235]}, {"type": "text", "bbox": [110, 239, 500, 281], "lines": [{"bbox": [127, 241, 499, 254], "spans": [{"bbox": [127, 241, 433, 254], "score": 1.0, "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "type": "text"}, {"bbox": [433, 243, 443, 251], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [443, 241, 492, 254], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [493, 243, 499, 251], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}], "index": 7}, {"bbox": [110, 255, 500, 269], "spans": [{"bbox": [110, 255, 400, 269], "score": 1.0, "content": "vertices with different labels and is composed by exactly ", "type": "text"}, {"bbox": [401, 257, 407, 266], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [407, 255, 447, 269], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [447, 257, 455, 266], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [455, 255, 500, 269], "score": 1.0, "content": " (in fact,", "type": "text"}], "index": 8}, {"bbox": [110, 270, 384, 284], "spans": [{"bbox": [110, 272, 123, 280], "score": 0.85, "content": "\\mathit{2a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [123, 270, 208, 284], "score": 1.0, "content": " horizontal arcs, ", "type": "text"}, {"bbox": [208, 271, 213, 280], "score": 0.89, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [214, 270, 304, 284], "score": 1.0, "content": " oblique arcs and ", "type": "text"}, {"bbox": [305, 275, 310, 280], "score": 0.89, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [311, 270, 384, 284], "score": 1.0, "content": " vertical arcs).", "type": "text"}], "index": 9}], "index": 8, "bbox_fs": [110, 241, 500, 284]}, {"type": "text", "bbox": [110, 282, 500, 325], "lines": [{"bbox": [127, 285, 499, 297], "spans": [{"bbox": [127, 285, 382, 297], "score": 1.0, "content": "An important consequence of point a) of Lemma ", "type": "text"}, {"bbox": [382, 286, 388, 294], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [388, 285, 443, 297], "score": 1.0, "content": " is that, if ", "type": "text"}, {"bbox": [443, 289, 450, 294], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [451, 285, 499, 297], "score": 1.0, "content": " is an ad-", "type": "text"}], "index": 10}, {"bbox": [110, 298, 500, 312], "spans": [{"bbox": [110, 298, 426, 312], "score": 1.0, "content": "missible 6-tuple, the presentation of the fundamental group of ", "type": "text"}, {"bbox": [427, 299, 456, 312], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [456, 298, 500, 312], "score": 1.0, "content": " induced", "type": "text"}], "index": 11}, {"bbox": [110, 313, 317, 326], "spans": [{"bbox": [110, 313, 243, 326], "score": 1.0, "content": "by the Heegaard diagram ", "type": "text"}, {"bbox": [244, 314, 271, 326], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [271, 313, 317, 326], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [110, 285, 500, 326]}, {"type": "text", "bbox": [109, 326, 500, 484], "lines": [{"bbox": [127, 327, 499, 341], "spans": [{"bbox": [127, 327, 206, 341], "score": 1.0, "content": "To see this, let ", "type": "text"}, {"bbox": [206, 333, 213, 338], "score": 0.88, "content": "\\upsilon", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 327, 403, 341], "score": 1.0, "content": " be the vertex belonging to the cycle ", "type": "text"}, {"bbox": [403, 329, 417, 340], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [417, 327, 499, 341], "score": 1.0, "content": " and labelled by", "type": "text"}], "index": 13}, {"bbox": [110, 342, 500, 356], "spans": [{"bbox": [110, 344, 159, 353], "score": 0.93, "content": "a+b+1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [159, 342, 222, 356], "score": 1.0, "content": "; denote by ", "type": "text"}, {"bbox": [222, 344, 237, 354], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [237, 342, 308, 356], "score": 1.0, "content": " the curve of ", "type": "text"}, {"bbox": [309, 344, 318, 352], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [319, 342, 380, 356], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [381, 347, 387, 352], "score": 0.89, "content": "v", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [387, 342, 432, 356], "score": 1.0, "content": " and by ", "type": "text"}, {"bbox": [433, 343, 442, 352], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [442, 342, 500, 356], "score": 1.0, "content": " the vertex", "type": "text"}], "index": 14}, {"bbox": [110, 356, 500, 370], "spans": [{"bbox": [110, 356, 124, 370], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [124, 358, 137, 370], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [137, 356, 232, 370], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [232, 361, 239, 367], "score": 0.89, "content": "v", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [239, 356, 327, 370], "score": 1.0, "content": ". Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18, "bbox_fs": [108, 327, 501, 486]}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". The letters of ", "type": "text"}, {"bbox": [232, 520, 241, 525], "score": 0.88, "content": "w", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [241, 515, 500, 529], "score": 1.0, "content": " are in one-to-one correspondence with the oriented", "type": "text"}], "index": 26}, {"bbox": [110, 530, 500, 543], "spans": [{"bbox": [110, 530, 134, 543], "score": 1.0, "content": "arcs ", "type": "text"}, {"bbox": [135, 534, 146, 542], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [146, 530, 298, 543], "score": 1.0, "content": "; more precisely, the letter of ", "type": "text"}, {"bbox": [298, 534, 307, 540], "score": 0.89, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [307, 530, 401, 543], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [401, 534, 412, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [412, 530, 428, 543], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [428, 534, 438, 542], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [438, 530, 453, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [453, 534, 464, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [464, 530, 500, 543], "score": 1.0, "content": " comes", "type": "text"}], "index": 27}, {"bbox": [109, 543, 501, 560], "spans": [{"bbox": [109, 543, 209, 560], "score": 1.0, "content": "out from the cycle ", "type": "text"}, {"bbox": [209, 545, 221, 558], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [222, 543, 260, 560], "score": 1.0, "content": " and is ", "type": "text"}, {"bbox": [261, 544, 279, 558], "score": 0.95, "content": "x_{i}^{-1}", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [279, 543, 293, 560], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [294, 549, 304, 556], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [305, 543, 442, 560], "score": 1.0, "content": " comes out from the cycle ", "type": "text"}, {"bbox": [443, 545, 465, 558], "score": 0.93, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [465, 543, 501, 560], "score": 1.0, "content": ". Note", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 573], "spans": [{"bbox": [109, 558, 181, 573], "score": 1.0, "content": "that the word ", "type": "text"}, {"bbox": [181, 559, 221, 572], "score": 0.94, "content": "\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [221, 558, 352, 573], "score": 1.0, "content": " in the cyclic presentation ", "type": "text"}, {"bbox": [352, 560, 385, 572], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [386, 558, 500, 573], "score": 1.0, "content": " is obtained by reading", "type": "text"}], "index": 29}, {"bbox": [109, 572, 499, 588], "spans": [{"bbox": [109, 572, 160, 588], "score": 1.0, "content": "the cycle ", "type": "text"}, {"bbox": [160, 575, 175, 586], "score": 0.92, "content": "D_{k}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [176, 572, 344, 588], "score": 1.0, "content": " along the given orientation, for ", "type": "text"}, {"bbox": [345, 575, 399, 585], "score": 0.92, "content": "1\\leq k\\leq n", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [400, 572, 499, 588], "score": 1.0, "content": " (roughly speaking,", "type": "text"}], "index": 30}, {"bbox": [109, 587, 398, 602], "spans": [{"bbox": [109, 587, 207, 602], "score": 1.0, "content": "the automorphism ", "type": "text"}, {"bbox": [208, 589, 219, 600], "score": 0.92, "content": "\\theta_{n}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [219, 587, 377, 602], "score": 1.0, "content": " is “geometrically” realized by ", "type": "text"}, {"bbox": [377, 592, 389, 600], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [389, 587, 398, 602], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 27.5, "bbox_fs": [109, 486, 501, 602]}, {"type": "text", "bbox": [109, 601, 500, 672], "lines": [{"bbox": [127, 601, 499, 616], "spans": [{"bbox": [127, 601, 336, 616], "score": 1.0, "content": "This proves that each admissible 6-tuple ", "type": "text"}, {"bbox": [336, 607, 343, 612], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [344, 601, 499, 616], "score": 1.0, "content": " uniquely defines, via the asso-", "type": "text"}], "index": 32}, {"bbox": [110, 617, 499, 630], "spans": [{"bbox": [110, 617, 242, 630], "score": 1.0, "content": "ciated Heegaard diagram ", "type": "text"}, {"bbox": [243, 617, 270, 630], "score": 0.94, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [270, 617, 316, 630], "score": 1.0, "content": ", a word ", "type": "text"}, {"bbox": [316, 617, 366, 630], "score": 0.95, "content": "w=w(\\sigma)", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [367, 617, 499, 630], "score": 1.0, "content": " and a cyclic presentation", "type": "text"}], "index": 33}, {"bbox": [110, 631, 500, 644], "spans": [{"bbox": [110, 632, 143, 644], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [144, 631, 434, 644], "score": 1.0, "content": " for the fundamental group of the Dunwoody manifold ", "type": "text"}, {"bbox": [434, 632, 463, 644], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [464, 631, 500, 644], "score": 1.0, "content": ". Note", "type": "text"}], "index": 34}, {"bbox": [110, 645, 499, 659], "spans": [{"bbox": [110, 645, 348, 659], "score": 1.0, "content": "that the sequence of the exponents in the word ", "type": "text"}, {"bbox": [348, 646, 373, 658], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [373, 645, 499, 659], "score": 1.0, "content": ", and hence its exponent-", "type": "text"}], "index": 35}, {"bbox": [109, 659, 359, 675], "spans": [{"bbox": [109, 659, 134, 675], "score": 1.0, "content": "sum ", "type": "text"}, {"bbox": [135, 664, 159, 674], "score": 0.92, "content": "\\varepsilon_{w(\\sigma)}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [160, 659, 316, 675], "score": 1.0, "content": ", only depends on the integers ", "type": "text"}, {"bbox": [317, 661, 354, 672], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [355, 659, 359, 675], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34, "bbox_fs": [109, 601, 500, 675]}]}
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[{"type": "text", "bbox": [110, 125, 500, 169], "content": "", "index": 0}, {"type": "text", "bbox": [110, 174, 501, 232], "content": "Remark 2. More generally, given two positive integer and such that divides , if is admissible, then is admissible too. Moreover, the Heegaard diagram is the quotient of respect to the action of a cyclic group of order .", "index": 1}, {"type": "text", "bbox": [110, 239, 500, 281], "content": "It is easy to see that, for admissible 6-tuples, each cycle in contains vertices with different labels and is composed by exactly arcs of (in fact, horizontal arcs, oblique arcs and vertical arcs).", "index": 2}, {"type": "text", "bbox": [110, 282, 500, 325], "content": "An important consequence of point a) of Lemma is that, if is an ad- missible 6-tuple, the presentation of the fundamental group of induced by the Heegaard diagram is cyclic.", "index": 3}, {"type": "text", "bbox": [109, 326, 500, 484], "content": "To see this, let be the vertex belonging to the cycle and labelled by ; denote by the curve of containing and by the vertex of corresponding to . Orient the arc of the graph containing so that is its first endpoint and orient the curve in accordance with the orientation of this arc. Now, set , for each ; the orientation on induces, via , an orientation also on these curves. Moreover, these orientation on the cycles of induce an orientation on the arcs of the graph belonging to . By orienting the arcs of and in accordance with the fixed orientations of the cycles and , the graph becomes an oriented graph, whose orientation is invariant under the action of the group . Let us define to be canonical this orientation of .", "index": 4}, {"type": "text", "bbox": [109, 485, 501, 599], "content": "Let now be the word obtained by reading the oriented arcs of corresponding to the oriented cycle , starting from the vertex . The letters of are in one-to-one correspondence with the oriented arcs ; more precisely, the letter of corresponding to is if comes out from the cycle and is if comes out from the cycle . Note that the word in the cyclic presentation is obtained by reading the cycle along the given orientation, for (roughly speaking, the automorphism is “geometrically” realized by ).", "index": 5}, {"type": "text", "bbox": [109, 601, 500, 672], "content": "This proves that each admissible 6-tuple uniquely defines, via the asso- ciated Heegaard diagram , a word and a cyclic presentation for the fundamental group of the Dunwoody manifold . Note that the sequence of the exponents in the word , and hence its exponent- sum , only depends on the integers .", "index": 6}]
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[{"bbox": [109, 176, 500, 191], "content": "Remark 2. More generally, given two positive integer and such that", "parent_index": 1, "line_index": 0}, {"bbox": [110, 191, 500, 205], "content": "divides , if is admissible, then is admissible", "parent_index": 1, "line_index": 1}, {"bbox": [109, 206, 502, 220], "content": "too. Moreover, the Heegaard diagram is the quotient of", "parent_index": 1, "line_index": 2}, {"bbox": [110, 220, 462, 235], "content": "respect to the action of a cyclic group of order .", "parent_index": 1, "line_index": 3}, {"bbox": [127, 241, 499, 254], "content": "It is easy to see that, for admissible 6-tuples, each cycle in contains", "parent_index": 2, "line_index": 0}, {"bbox": [110, 255, 500, 269], "content": "vertices with different labels and is composed by exactly arcs of (in fact,", "parent_index": 2, "line_index": 1}, {"bbox": [110, 270, 384, 284], "content": "horizontal arcs, oblique arcs and vertical arcs).", "parent_index": 2, "line_index": 2}, {"bbox": [127, 285, 499, 297], "content": "An important consequence of point a) of Lemma is that, if is an ad-", "parent_index": 3, "line_index": 0}, {"bbox": [110, 298, 500, 312], "content": "missible 6-tuple, the presentation of the fundamental group of induced", "parent_index": 3, "line_index": 1}, {"bbox": [110, 313, 317, 326], "content": "by the Heegaard diagram is cyclic.", "parent_index": 3, "line_index": 2}, {"bbox": [127, 327, 499, 341], "content": "To see this, let be the vertex belonging to the cycle and labelled by", "parent_index": 4, "line_index": 0}, {"bbox": [110, 342, 500, 356], "content": "; denote by the curve of containing and by the vertex", "parent_index": 4, "line_index": 1}, {"bbox": [110, 356, 500, 370], "content": "of corresponding to . Orient the arc of the graph containing", "parent_index": 4, "line_index": 2}, {"bbox": [110, 370, 500, 385], "content": "so that is its first endpoint and orient the curve in accordance with", "parent_index": 4, "line_index": 3}, {"bbox": [108, 385, 501, 401], "content": "the orientation of this arc. Now, set , for each ;", "parent_index": 4, "line_index": 4}, {"bbox": [109, 399, 501, 414], "content": "the orientation on induces, via , an orientation also on these curves.", "parent_index": 4, "line_index": 5}, {"bbox": [110, 415, 499, 427], "content": "Moreover, these orientation on the cycles of induce an orientation on the", "parent_index": 4, "line_index": 6}, {"bbox": [109, 428, 501, 442], "content": "arcs of the graph belonging to . By orienting the arcs of and in", "parent_index": 4, "line_index": 7}, {"bbox": [109, 443, 499, 458], "content": "accordance with the fixed orientations of the cycles and , the graph", "parent_index": 4, "line_index": 8}, {"bbox": [109, 457, 500, 471], "content": "becomes an oriented graph, whose orientation is invariant under the action", "parent_index": 4, "line_index": 9}, {"bbox": [110, 471, 459, 486], "content": "of the group . Let us define to be canonical this orientation of .", "parent_index": 4, "line_index": 10}, {"bbox": [126, 486, 500, 500], "content": "Let now be the word obtained by reading the oriented arcs", "parent_index": 5, "line_index": 0}, {"bbox": [110, 500, 501, 516], "content": "of corresponding to the oriented cycle , starting from the", "parent_index": 5, "line_index": 1}, {"bbox": [110, 515, 500, 529], "content": "vertex . The letters of are in one-to-one correspondence with the oriented", "parent_index": 5, "line_index": 2}, {"bbox": [110, 530, 500, 543], "content": "arcs ; more precisely, the letter of corresponding to is if comes", "parent_index": 5, "line_index": 3}, {"bbox": [109, 543, 501, 560], "content": "out from the cycle and is if comes out from the cycle . Note", "parent_index": 5, "line_index": 4}, {"bbox": [109, 558, 500, 573], "content": "that the word in the cyclic presentation is obtained by reading", "parent_index": 5, "line_index": 5}, {"bbox": [109, 572, 499, 588], "content": "the cycle along the given orientation, for (roughly speaking,", "parent_index": 5, "line_index": 6}, {"bbox": [109, 587, 398, 602], "content": "the automorphism is “geometrically” realized by ).", "parent_index": 5, "line_index": 7}, {"bbox": [127, 601, 499, 616], "content": "This proves that each admissible 6-tuple uniquely defines, via the asso-", "parent_index": 6, "line_index": 0}, {"bbox": [110, 617, 499, 630], "content": "ciated Heegaard diagram , a word and a cyclic presentation", "parent_index": 6, "line_index": 1}, {"bbox": [110, 631, 500, 644], "content": "for the fundamental group of the Dunwoody manifold . Note", "parent_index": 6, "line_index": 2}, {"bbox": [110, 645, 499, 659], "content": "that the sequence of the exponents in the word , and hence its exponent-", "parent_index": 6, "line_index": 3}, {"bbox": [109, 659, 359, 675], "content": "sum , only depends on the integers .", "parent_index": 6, "line_index": 4}]
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[]
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Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram.
Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to:
i) $\mathbf{S^{3}}$ , if $\varepsilon_{w}=\pm1$ ;
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $\varepsilon_{w}=0$ ;
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|\varepsilon_{w}|$ , if $\left|\varepsilon_{w}\right|>1$ .
Proof. From $n=1$ we obtain $w\in F_{1}\cong\mathbf{Z}\cong<x|\emptyset>$ . Thus, $\pi_{1}(M)\cong$ $G_{1}(w)\cong<x|x^{\varepsilon_{w}}>\cong{\mathbf{Z}}_{|\varepsilon_{w}|}$ .
Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\mathbf{S^{3}}$ , $\mathbf{S^{1}\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB.
Let us see now how the admissibility conditions for the 6-tuples of $\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\Gamma^{\prime}$ , belonging to the curve $D_{1}\in\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\sigma\in S$ :
(i’) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ;
(ii’) the vertices of the cycle $D_{1}$ have different labels.
It is easy to see that, if a 6-tuple $\sigma\in S$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $\sigma\,\in\,S$ satisfies (i’) and (ii’), then the curves $\rho_{n}^{k-1}(D_{1})\,\in\,\mathcal{D}$ , with $k\,=\,1,\ldots,n$ , which are all different from each other, are precisely the curves of $\mathcal{D}$ . Thus, $\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\rho_{n}$ . However, this does not imply that $\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii’) always holds, while condition (i’) holds if and only if $\mathcal{D}$ has a unique cycle.
If a 6-tuple satisfies property (i’), then $\mathcal{G}_{n}$ acts transitively (not necessarily simply) on $\mathcal{D}$ , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of $\mathcal{D}$ and on the graph $\Gamma$ , by extending, via $\rho_{n}$ , the orientation of $D_{1}$ to the other cycles of $\mathcal{D}$ .
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<p data-bbox="110 125 501 154">Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram. </p>
<p data-bbox="109 166 501 210">Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to: </p>
<p data-bbox="126 211 385 255">i) $\mathbf{S^{3}}$ , if $\varepsilon_{w}=\pm1$ ;
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $\varepsilon_{w}=0$ ;
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|\varepsilon_{w}|$ , if $\left|\varepsilon_{w}\right|>1$ . </p>
<p data-bbox="109 266 500 296">Proof. From $n=1$ we obtain $w\in F_{1}\cong\mathbf{Z}\cong<x|\emptyset>$ . Thus, $\pi_{1}(M)\cong$ $G_{1}(w)\cong<x|x^{\varepsilon_{w}}>\cong{\mathbf{Z}}_{|\varepsilon_{w}|}$ . </p>
<p data-bbox="109 300 500 403">Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\mathbf{S^{3}}$ , $\mathbf{S^{1}\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB. </p>
<p data-bbox="110 409 500 452">Let us see now how the admissibility conditions for the 6-tuples of $\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\Gamma^{\prime}$ , belonging to the curve $D_{1}\in\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\sigma\in S$ : </p>
<p data-bbox="116 461 499 491">(i’) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ; </p>
<p data-bbox="114 501 388 515">(ii’) the vertices of the cycle $D_{1}$ have different labels. </p>
<p data-bbox="109 525 500 641">It is easy to see that, if a 6-tuple $\sigma\in S$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $\sigma\,\in\,S$ satisfies (i’) and (ii’), then the curves $\rho_{n}^{k-1}(D_{1})\,\in\,\mathcal{D}$ , with $k\,=\,1,\ldots,n$ , which are all different from each other, are precisely the curves of $\mathcal{D}$ . Thus, $\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\rho_{n}$ . However, this does not imply that $\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii’) always holds, while condition (i’) holds if and only if $\mathcal{D}$ has a unique cycle. </p>
<p data-bbox="109 642 501 670">If a 6-tuple satisfies property (i’), then $\mathcal{G}_{n}$ acts transitively (not necessarily simply) on $\mathcal{D}$ , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of $\mathcal{D}$ and on the graph $\Gamma$ , by extending, via $\rho_{n}$ , the orientation of $D_{1}$ to the other cycles of $\mathcal{D}$ . </p>
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0003042v1
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[{"type": "text", "text": "Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram. ", "page_idx": 7}, {"type": "text", "text": "Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to: ", "page_idx": 7}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $\\varepsilon_{w}=\\pm1$ ; \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $\\varepsilon_{w}=0$ ; \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=|\\varepsilon_{w}|$ , if $\\left|\\varepsilon_{w}\\right|>1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. From $n=1$ we obtain $w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>$ . Thus, $\\pi_{1}(M)\\cong$ $G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}$ . ", "page_idx": 7}, {"type": "text", "text": "Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}\\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB. ", "page_idx": 7}, {"type": "text", "text": "Let us see now how the admissibility conditions for the 6-tuples of $\\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\\Gamma^{\\prime}$ , belonging to the curve $D_{1}\\in\\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\\sigma\\in S$ : ", "page_idx": 7}, {"type": "text", "text": "(i’) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ; ", "page_idx": 7}, {"type": "text", "text": "(ii’) the vertices of the cycle $D_{1}$ have different labels. ", "page_idx": 7}, {"type": "text", "text": "It is easy to see that, if a 6-tuple $\\sigma\\in S$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $\\sigma\\,\\in\\,S$ satisfies (i’) and (ii’), then the curves $\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}$ , with $k\\,=\\,1,\\ldots,n$ , which are all different from each other, are precisely the curves of $\\mathcal{D}$ . Thus, $\\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\\rho_{n}$ . However, this does not imply that $\\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii’) always holds, while condition (i’) holds if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 7}, {"type": "text", "text": "If a 6-tuple satisfies property (i’), then $\\mathcal{G}_{n}$ acts transitively (not necessarily simply) on $\\mathcal{D}$ , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of $\\mathcal{D}$ and on the graph $\\Gamma$ , by extending, via $\\rho_{n}$ , the orientation of $D_{1}$ to the other cycles of $\\mathcal{D}$ . ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=|\\varepsilon_{w}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left|\\varepsilon_{w}\\right|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i’) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii’) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i’)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii’). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i’) and (ii’), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i’) and (ii’),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii’)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i’) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i’), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}], "index": 31.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [111, 127, 499, 156]}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [110, 169, 500, 212]}, {"type": "list", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5, "is_list_end_line": true}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=|\\varepsilon_{w}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left|\\varepsilon_{w}\\right|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}], "index": 6, "bbox_fs": [127, 212, 384, 255]}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5, "bbox_fs": [110, 269, 501, 300]}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13, "bbox_fs": [109, 304, 501, 405]}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [109, 411, 501, 454]}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i’) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [118, 464, 501, 492]}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii’) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [115, 503, 388, 516]}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i’)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii’). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i’) and (ii’), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i’) and (ii’),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii’)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i’) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5, "bbox_fs": [108, 527, 501, 644]}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i’), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}, {"bbox": [109, 127, 499, 142], "spans": [{"bbox": [109, 127, 306, 142], "score": 1.0, "content": "said to be canonical) on the cycles of ", "type": "text", "cross_page": true}, {"bbox": [307, 130, 316, 138], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [317, 127, 414, 142], "score": 1.0, "content": " and on the graph ", "type": "text", "cross_page": true}, {"bbox": [414, 129, 422, 138], "score": 0.87, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [422, 127, 499, 142], "score": 1.0, "content": ", by extending,", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 143, 385, 155], "spans": [{"bbox": [110, 143, 129, 155], "score": 1.0, "content": "via ", "type": "text", "cross_page": true}, {"bbox": [129, 147, 141, 155], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12, "cross_page": true}, {"bbox": [141, 143, 241, 155], "score": 1.0, "content": ", the orientation of ", "type": "text", "cross_page": true}, {"bbox": [241, 144, 255, 154], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [256, 143, 371, 155], "score": 1.0, "content": " to the other cycles of ", "type": "text", "cross_page": true}, {"bbox": [371, 144, 381, 153], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [381, 143, 385, 155], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 1}], "index": 31.5, "bbox_fs": [110, 642, 500, 673]}]}
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[{"type": "text", "bbox": [110, 125, 501, 154], "content": "Let us consider now the Dunwoody manifolds with (and hence ), which arises from a genus one Heegaard diagram.", "index": 0}, {"type": "text", "bbox": [109, 166, 501, 210], "content": "Proposition 2 Let be an admissible 6-tuple and let be the associated word. Then the Dunwoody manifold is homeomorphic to:", "index": 1}, {"type": "list", "bbox": [126, 211, 385, 255], "content": "", "index": 2}, {"type": "text", "bbox": [109, 266, 500, 296], "content": "Proof. From we obtain . Thus, .", "index": 3}, {"type": "text", "bbox": [109, 300, 500, 403], "content": "Example 1. The Dunwoody manifolds , and , with coprime, are homeomorphic to , and to the lens space , respectively. Moreover, all lens spaces also arise with ; in fact, for each , is homeomorphic with the lens space , if and are coprime, since it is easy to see that can be transformed into the canonical genus one Heegaard diagram of by Singer moves of type IB.", "index": 4}, {"type": "text", "bbox": [110, 409, 500, 452], "content": "Let us see now how the admissibility conditions for the 6-tuples of can be given in terms of labelling of the vertices of , belonging to the curve . With this aim, consider the following properties for a 6-tuple :", "index": 5}, {"type": "text", "bbox": [116, 461, 499, 491], "content": "(i’) the set of the labels of the vertices belonging to the cycle is the set of all integers from 1 to ;", "index": 6}, {"type": "text", "bbox": [114, 501, 388, 515], "content": "(ii’) the vertices of the cycle have different labels.", "index": 7}, {"type": "text", "bbox": [109, 525, 500, 641], "content": "It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then the curves , with , which are all different from each other, are precisely the curves of . Thus, has exactly curves and they are cyclically permutated by . However, this does not imply that is admissible; for example, the 6-tuple satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for , property (ii’) always holds, while condition (i’) holds if and only if has a unique cycle.", "index": 8}, {"type": "text", "bbox": [109, 642, 501, 670], "content": "If a 6-tuple satisfies property (i’), then acts transitively (not necessarily simply) on , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of and on the graph , by extending, via , the orientation of to the other cycles of .", "index": 9}]
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[{"bbox": [126, 127, 499, 142], "content": "Let us consider now the Dunwoody manifolds with", "parent_index": 0, "line_index": 0}, {"bbox": [111, 142, 462, 156], "content": "(and hence ), which arises from a genus one Heegaard diagram.", "parent_index": 0, "line_index": 1}, {"bbox": [110, 169, 500, 183], "content": "Proposition 2 Let be an admissible 6-tuple and let", "parent_index": 1, "line_index": 0}, {"bbox": [110, 183, 500, 198], "content": "be the associated word. Then the Dunwoody manifold", "parent_index": 1, "line_index": 1}, {"bbox": [110, 198, 300, 212], "content": "is homeomorphic to:", "parent_index": 1, "line_index": 2}, {"bbox": [127, 212, 221, 227], "content": "i) , if ;", "parent_index": 2, "line_index": 0}, {"bbox": [127, 226, 243, 241], "content": "ii) , if ;", "parent_index": 2, "line_index": 1}, {"bbox": [127, 241, 384, 255], "content": "iii) a lens space with , if .", "parent_index": 2, "line_index": 2}, {"bbox": [126, 269, 501, 284], "content": "Proof. From we obtain . Thus,", "parent_index": 3, "line_index": 0}, {"bbox": [110, 281, 261, 300], "content": ".", "parent_index": 3, "line_index": 1}, {"bbox": [110, 304, 499, 318], "content": "Example 1. The Dunwoody manifolds ,", "parent_index": 4, "line_index": 0}, {"bbox": [109, 318, 499, 333], "content": "and , with coprime, are homeomorphic to ,", "parent_index": 4, "line_index": 1}, {"bbox": [110, 333, 500, 347], "content": "and to the lens space , respectively. Moreover, all lens spaces also arise", "parent_index": 4, "line_index": 2}, {"bbox": [110, 347, 499, 362], "content": "with ; in fact, for each , is homeomorphic with", "parent_index": 4, "line_index": 3}, {"bbox": [110, 362, 501, 376], "content": "the lens space , if and are coprime, since it is easy to see that", "parent_index": 4, "line_index": 4}, {"bbox": [110, 375, 501, 390], "content": "can be transformed into the canonical genus one Heegaard", "parent_index": 4, "line_index": 5}, {"bbox": [109, 390, 349, 405], "content": "diagram of by Singer moves of type IB.", "parent_index": 4, "line_index": 6}, {"bbox": [126, 411, 499, 425], "content": "Let us see now how the admissibility conditions for the 6-tuples of", "parent_index": 5, "line_index": 0}, {"bbox": [109, 426, 501, 440], "content": "can be given in terms of labelling of the vertices of , belonging to the curve", "parent_index": 5, "line_index": 1}, {"bbox": [110, 440, 500, 454], "content": ". With this aim, consider the following properties for a 6-tuple :", "parent_index": 5, "line_index": 2}, {"bbox": [118, 464, 501, 479], "content": "(i’) the set of the labels of the vertices belonging to the cycle is the set", "parent_index": 6, "line_index": 0}, {"bbox": [139, 478, 275, 492], "content": "of all integers from 1 to ;", "parent_index": 6, "line_index": 1}, {"bbox": [115, 503, 388, 516], "content": "(ii’) the vertices of the cycle have different labels.", "parent_index": 7, "line_index": 0}, {"bbox": [126, 527, 499, 542], "content": "It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’)", "parent_index": 8, "line_index": 0}, {"bbox": [110, 542, 500, 556], "content": "and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then", "parent_index": 8, "line_index": 1}, {"bbox": [109, 556, 501, 572], "content": "the curves , with , which are all different from", "parent_index": 8, "line_index": 2}, {"bbox": [109, 571, 501, 586], "content": "each other, are precisely the curves of . Thus, has exactly curves and", "parent_index": 8, "line_index": 3}, {"bbox": [109, 585, 499, 600], "content": "they are cyclically permutated by . However, this does not imply that", "parent_index": 8, "line_index": 4}, {"bbox": [109, 600, 499, 614], "content": "is admissible; for example, the 6-tuple satisfies (i’) and (ii’),", "parent_index": 8, "line_index": 5}, {"bbox": [108, 614, 499, 629], "content": "but it is not admissible (see Remark 1). Note that, for , property (ii’)", "parent_index": 8, "line_index": 6}, {"bbox": [110, 629, 494, 644], "content": "always holds, while condition (i’) holds if and only if has a unique cycle.", "parent_index": 8, "line_index": 7}, {"bbox": [126, 642, 499, 659], "content": "If a 6-tuple satisfies property (i’), then acts transitively (not necessarily", "parent_index": 9, "line_index": 0}, {"bbox": [110, 657, 500, 673], "content": "simply) on , and hence it is possible to induce an orientation (which is still", "parent_index": 9, "line_index": 1}, {"bbox": [109, 127, 499, 142], "content": "said to be canonical) on the cycles of and on the graph , by extending,", "parent_index": 9, "line_index": 2}, {"bbox": [110, 143, 385, 155], "content": "via , the orientation of to the other cycles of .", "parent_index": 9, "line_index": 3}]
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[]
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[{"bbox": [360, 129, 441, 141], "content": "M(a,b,c,n,r,s)", "parent_index": 0, "subtype": "inline"}, {"bbox": [470, 130, 499, 138], "content": "n=1", "parent_index": 0, "subtype": "inline"}, {"bbox": [170, 144, 198, 153], "content": "s=0", "parent_index": 0, "subtype": "inline"}, {"bbox": [218, 169, 286, 183], "content": "(a,b,c,1,r,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [472, 170, 500, 182], "content": "w\\ =", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 183, 188, 197], "content": "w(a,b,c,1,r,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 198, 191, 212], "content": "M(a,b,c,1,r,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [139, 212, 154, 225], "content": "\\mathbf{S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [171, 213, 216, 225], "content": "\\varepsilon_{w}=\\pm1", "parent_index": 2, "subtype": "inline"}, {"bbox": [144, 226, 185, 239], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [203, 227, 238, 240], "content": "\\varepsilon_{w}=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [212, 241, 249, 255], "content": "L(\\alpha,\\beta)", "parent_index": 2, "subtype": "inline"}, {"bbox": [278, 241, 321, 255], "content": "\\alpha=|\\varepsilon_{w}|", "parent_index": 2, "subtype": "inline"}, {"bbox": [340, 243, 381, 255], "content": "\\left|\\varepsilon_{w}\\right|>1", "parent_index": 2, "subtype": "inline"}, {"bbox": [201, 270, 232, 280], "content": "n=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [290, 270, 409, 282], "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>", "parent_index": 3, "subtype": "inline"}, {"bbox": [452, 270, 501, 283], "content": "\\pi_{1}(M)\\cong", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 282, 243, 298], "content": "G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}", "parent_index": 3, "subtype": "inline"}, {"bbox": [324, 304, 408, 317], "content": "M(0,0,1,1,0,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [416, 305, 499, 317], "content": "M(1,0,0,1,1,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [134, 318, 216, 332], "content": "M(0,0,c,1,r,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [251, 323, 267, 331], "content": "c,r", "parent_index": 4, "subtype": "inline"}, {"bbox": [436, 319, 449, 329], "content": "\\mathbf{S^{3}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [457, 319, 499, 330], "content": "\\mathbf{S^{1}\\times S^{2}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [218, 333, 252, 347], "content": "L(c,r)", "parent_index": 4, "subtype": "inline"}, {"bbox": [137, 349, 165, 360], "content": "a\\neq0", "parent_index": 4, "subtype": "inline"}, {"bbox": [262, 349, 291, 358], "content": "a>0", "parent_index": 4, "subtype": "inline"}, {"bbox": [298, 348, 381, 361], "content": "M(a,0,c,1,a,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [189, 363, 222, 375], "content": "L(c,a)", "parent_index": 4, "subtype": "inline"}, {"bbox": [243, 367, 249, 372], "content": "a", "parent_index": 4, "subtype": "inline"}, {"bbox": [279, 367, 284, 372], "content": "c", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 377, 191, 389], "content": "H(a,0,c,1,a,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [169, 392, 202, 404], "content": "L(c,a)", "parent_index": 4, "subtype": "inline"}, {"bbox": [490, 413, 499, 421], "content": "\\boldsymbol{S}", "parent_index": 5, "subtype": "inline"}, {"bbox": [369, 427, 379, 436], "content": "\\Gamma^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 442, 149, 452], "content": "D_{1}\\in\\mathcal{D}", "parent_index": 5, "subtype": "inline"}, {"bbox": [466, 442, 496, 451], "content": "\\sigma\\in S", "parent_index": 5, "subtype": "inline"}, {"bbox": [434, 466, 449, 477], "content": "D_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [264, 480, 271, 489], "content": "d", "parent_index": 6, "subtype": "inline"}, {"bbox": [264, 505, 279, 515], "content": "D_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [298, 529, 327, 538], "content": "\\sigma\\in S", "parent_index": 8, "subtype": "inline"}, {"bbox": [325, 544, 357, 553], "content": "\\sigma\\,\\in\\,S", "parent_index": 8, "subtype": "inline"}, {"bbox": [168, 557, 242, 570], "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "parent_index": 8, "subtype": "inline"}, {"bbox": [279, 558, 347, 569], "content": "k\\,=\\,1,\\ldots,n", "parent_index": 8, "subtype": "inline"}, {"bbox": [307, 573, 317, 581], "content": "\\mathcal{D}", "parent_index": 8, "subtype": "inline"}, {"bbox": [358, 573, 368, 581], "content": "\\mathcal{D}", "parent_index": 8, "subtype": "inline"}, {"bbox": [433, 576, 440, 581], "content": "n", "parent_index": 8, "subtype": "inline"}, {"bbox": [288, 590, 300, 598], "content": "\\rho_{n}", "parent_index": 8, "subtype": "inline"}, {"bbox": [492, 590, 499, 596], "content": "\\sigma", "parent_index": 8, "subtype": "inline"}, {"bbox": [314, 600, 385, 613], "content": "(1,0,2,1,2,0)", "parent_index": 8, "subtype": "inline"}, {"bbox": [396, 617, 425, 625], "content": "n=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [384, 631, 393, 639], "content": "\\mathcal{D}", "parent_index": 8, "subtype": "inline"}, {"bbox": [321, 645, 334, 655], "content": "\\mathcal{G}_{n}", "parent_index": 9, "subtype": "inline"}, {"bbox": [168, 660, 177, 668], "content": "\\mathcal{D}", "parent_index": 9, "subtype": "inline"}, {"bbox": [307, 130, 316, 138], "content": "\\mathcal{D}", "parent_index": 9, "subtype": "inline"}, {"bbox": [414, 129, 422, 138], "content": "\\Gamma", "parent_index": 9, "subtype": "inline"}, {"bbox": [129, 147, 141, 155], "content": "\\rho_{n}", "parent_index": 9, "subtype": "inline"}, {"bbox": [241, 144, 255, 154], "content": "D_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [371, 144, 381, 153], "content": "\\mathcal{D}", "parent_index": 9, "subtype": "inline"}]
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Property (i’) implies that the cycles of $\mathcal{D}$ naturally induce a cyclic permutation on the set $\mathcal{N}=\{1,\dotsc,d\}$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $v$ labelled $j$ , one sequentially meets $d$ vertices (whose labels are different from each other), and then a new vertex $\bar{v}^{\prime}$ labelled $j$ which can be different from $v$ . The sequence of the labellings of these $d$ consecutive vertices defines the cyclic permutation on $\mathcal{N}$ . Further, each cycle of $\mathcal{D}$ precisely contains $d^{\prime}=l d$ arcs, with $l\geq1$ , and $l=1$ if and only if the 6-tuple satisfies (ii’) too. Moreover, property (i’) is independent from the integers $n$ and $s$ ; hence, given two 6-tuples $\sigma\,=\,(a,b,c,n,r,s)$ and ${\boldsymbol{\sigma}}^{\prime}\,=\,(a,b,c,n^{\prime},r,s)$ , then $\sigma$ satisfies (i’) if and only if $\sigma^{\prime}$ satisfies (i’).
Let now $\sigma$ be a 6-tuple satisfying (i’) and suppose that $\Gamma$ is canonically oriented. An arc of $\Gamma$ belonging to $A$ is said to be of type I if it is oriented from a cycle of $\mathcal{C}^{\prime}$ to a cycle of $\mathcal{C^{\prime\prime}}$ , of type II if it is oriented from a cycle of $\mathcal{C^{\prime\prime}}$ to a cycle of $\mathcal{C}^{\prime}$ and of type III otherwise (it joins cycles of $\mathcal{C}^{\prime}$ or cycles of $\mathcal{C}^{\prime\prime})$ . Moreover, the arc is said to be of type I’ if it is oriented from a cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) to a cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ), of type II’ if it is oriented from a cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ) to a cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) and of type III’ otherwise (it joins $C_{i}^{\prime}$ with $C_{i}^{\prime\prime}$ ). Let $\Delta$ be the set of the first $d$ arcs of $D_{1}$ , following the canonical orientation, starting from the arc coming out from the vertex $v^{\prime}$ of $C_{1}^{\prime}$ labelled $a+b+1$ . Obviously, the set $\Delta$ contains all the arcs of $D_{1}$ if and only if the 6-tuple $\sigma$ also satisfies (ii’).
Now, denote by $p_{\sigma}^{\prime}$ (resp. $p_{\sigma}^{\prime\prime}$ ) the number of the arcs of type I (resp. of type II) of $\Delta$ and set $p_{\sigma}=p_{\sigma}^{\prime}-p_{\sigma}^{\prime\prime}$ . Similarly, denote by $q_{\sigma}^{\prime}$ (resp. $q_{\sigma}^{\prime\prime}$ ) the number of the arcs of type $\Gamma$ (resp. of type II’) of $\Delta$ and set $q_{\sigma}=q_{\sigma}^{\prime}-q_{\sigma}^{\prime\prime}$ . Note that $p_{\sigma}$ has the same parity of $b\!+\!c$ and $q_{\sigma}$ has the same parity of $2a+b$ and hence of $b$ . It is evident that $p_{\sigma}$ and $q_{\sigma}$ only depend on the integers $a,b,c,r$ .
The integers $p_{\sigma}$ and $q_{\sigma}$ give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle $D_{j}$ of $\mathcal{D}$ , starting from a vertex $v$ and let $C_{i}$ be the cycle of $\mathcal{C}$ containing $v$ . If $\bar{v}^{\prime}$ is the first vertex with the same label of $v$ and if $C_{i^{\prime}}$ is the cycle of $\mathcal{C}$ containing $\bar{v}^{\prime}$ , we have $\d j^{\prime}=\d j+\d q_{\sigma}+\d s p_{\sigma}$ . Thus, the cycle $D_{j}$ contains $d$ arcs if and only if $q_{\sigma}+s p_{\sigma}\equiv0$ (mod $n$ ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a
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<p data-bbox="109 155 500 313">Property (i’) implies that the cycles of $\mathcal{D}$ naturally induce a cyclic permutation on the set $\mathcal{N}=\{1,\dotsc,d\}$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $v$ labelled $j$ , one sequentially meets $d$ vertices (whose labels are different from each other), and then a new vertex $\bar{v}^{\prime}$ labelled $j$ which can be different from $v$ . The sequence of the labellings of these $d$ consecutive vertices defines the cyclic permutation on $\mathcal{N}$ . Further, each cycle of $\mathcal{D}$ precisely contains $d^{\prime}=l d$ arcs, with $l\geq1$ , and $l=1$ if and only if the 6-tuple satisfies (ii’) too. Moreover, property (i’) is independent from the integers $n$ and $s$ ; hence, given two 6-tuples $\sigma\,=\,(a,b,c,n,r,s)$ and ${\boldsymbol{\sigma}}^{\prime}\,=\,(a,b,c,n^{\prime},r,s)$ , then $\sigma$ satisfies (i’) if and only if $\sigma^{\prime}$ satisfies (i’). </p>
<p data-bbox="109 313 501 471">Let now $\sigma$ be a 6-tuple satisfying (i’) and suppose that $\Gamma$ is canonically oriented. An arc of $\Gamma$ belonging to $A$ is said to be of type I if it is oriented from a cycle of $\mathcal{C}^{\prime}$ to a cycle of $\mathcal{C^{\prime\prime}}$ , of type II if it is oriented from a cycle of $\mathcal{C^{\prime\prime}}$ to a cycle of $\mathcal{C}^{\prime}$ and of type III otherwise (it joins cycles of $\mathcal{C}^{\prime}$ or cycles of $\mathcal{C}^{\prime\prime})$ . Moreover, the arc is said to be of type I’ if it is oriented from a cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) to a cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ), of type II’ if it is oriented from a cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ) to a cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) and of type III’ otherwise (it joins $C_{i}^{\prime}$ with $C_{i}^{\prime\prime}$ ). Let $\Delta$ be the set of the first $d$ arcs of $D_{1}$ , following the canonical orientation, starting from the arc coming out from the vertex $v^{\prime}$ of $C_{1}^{\prime}$ labelled $a+b+1$ . Obviously, the set $\Delta$ contains all the arcs of $D_{1}$ if and only if the 6-tuple $\sigma$ also satisfies (ii’). </p>
<p data-bbox="110 473 500 559">Now, denote by $p_{\sigma}^{\prime}$ (resp. $p_{\sigma}^{\prime\prime}$ ) the number of the arcs of type I (resp. of type II) of $\Delta$ and set $p_{\sigma}=p_{\sigma}^{\prime}-p_{\sigma}^{\prime\prime}$ . Similarly, denote by $q_{\sigma}^{\prime}$ (resp. $q_{\sigma}^{\prime\prime}$ ) the number of the arcs of type $\Gamma$ (resp. of type II’) of $\Delta$ and set $q_{\sigma}=q_{\sigma}^{\prime}-q_{\sigma}^{\prime\prime}$ . Note that $p_{\sigma}$ has the same parity of $b\!+\!c$ and $q_{\sigma}$ has the same parity of $2a+b$ and hence of $b$ . It is evident that $p_{\sigma}$ and $q_{\sigma}$ only depend on the integers $a,b,c,r$ . </p>
<p data-bbox="109 559 500 675">The integers $p_{\sigma}$ and $q_{\sigma}$ give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle $D_{j}$ of $\mathcal{D}$ , starting from a vertex $v$ and let $C_{i}$ be the cycle of $\mathcal{C}$ containing $v$ . If $\bar{v}^{\prime}$ is the first vertex with the same label of $v$ and if $C_{i^{\prime}}$ is the cycle of $\mathcal{C}$ containing $\bar{v}^{\prime}$ , we have $\d j^{\prime}=\d j+\d q_{\sigma}+\d s p_{\sigma}$ . Thus, the cycle $D_{j}$ contains $d$ arcs if and only if $q_{\sigma}+s p_{\sigma}\equiv0$ (mod $n$ ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "Property (i’) implies that the cycles of $\\mathcal{D}$ naturally induce a cyclic permutation on the set $\\mathcal{N}=\\{1,\\dotsc,d\\}$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $v$ labelled $j$ , one sequentially meets $d$ vertices (whose labels are different from each other), and then a new vertex $\\bar{v}^{\\prime}$ labelled $j$ which can be different from $v$ . The sequence of the labellings of these $d$ consecutive vertices defines the cyclic permutation on $\\mathcal{N}$ . Further, each cycle of $\\mathcal{D}$ precisely contains $d^{\\prime}=l d$ arcs, with $l\\geq1$ , and $l=1$ if and only if the 6-tuple satisfies (ii’) too. Moreover, property (i’) is independent from the integers $n$ and $s$ ; hence, given two 6-tuples $\\sigma\\,=\\,(a,b,c,n,r,s)$ and ${\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)$ , then $\\sigma$ satisfies (i’) if and only if $\\sigma^{\\prime}$ satisfies (i’). ", "page_idx": 8}, {"type": "text", "text": "Let now $\\sigma$ be a 6-tuple satisfying (i’) and suppose that $\\Gamma$ is canonically oriented. An arc of $\\Gamma$ belonging to $A$ is said to be of type I if it is oriented from a cycle of $\\mathcal{C}^{\\prime}$ to a cycle of $\\mathcal{C^{\\prime\\prime}}$ , of type II if it is oriented from a cycle of $\\mathcal{C^{\\prime\\prime}}$ to a cycle of $\\mathcal{C}^{\\prime}$ and of type III otherwise (it joins cycles of $\\mathcal{C}^{\\prime}$ or cycles of $\\mathcal{C}^{\\prime\\prime})$ . Moreover, the arc is said to be of type I’ if it is oriented from a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) to a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ), of type II’ if it is oriented from a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) to a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) and of type III’ otherwise (it joins $C_{i}^{\\prime}$ with $C_{i}^{\\prime\\prime}$ ). Let $\\Delta$ be the set of the first $d$ arcs of $D_{1}$ , following the canonical orientation, starting from the arc coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ labelled $a+b+1$ . Obviously, the set $\\Delta$ contains all the arcs of $D_{1}$ if and only if the 6-tuple $\\sigma$ also satisfies (ii’). ", "page_idx": 8}, {"type": "text", "text": "Now, denote by $p_{\\sigma}^{\\prime}$ (resp. $p_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type I (resp. of type II) of $\\Delta$ and set $p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}$ . Similarly, denote by $q_{\\sigma}^{\\prime}$ (resp. $q_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type $\\Gamma$ (resp. of type II’) of $\\Delta$ and set $q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}$ . Note that $p_{\\sigma}$ has the same parity of $b\\!+\\!c$ and $q_{\\sigma}$ has the same parity of $2a+b$ and hence of $b$ . It is evident that $p_{\\sigma}$ and $q_{\\sigma}$ only depend on the integers $a,b,c,r$ . ", "page_idx": 8}, {"type": "text", "text": "The integers $p_{\\sigma}$ and $q_{\\sigma}$ give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle $D_{j}$ of $\\mathcal{D}$ , starting from a vertex $v$ and let $C_{i}$ be the cycle of $\\mathcal{C}$ containing $v$ . If $\\bar{v}^{\\prime}$ is the first vertex with the same label of $v$ and if $C_{i^{\\prime}}$ is the cycle of $\\mathcal{C}$ containing $\\bar{v}^{\\prime}$ , we have $\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}$ . Thus, the cycle $D_{j}$ contains $d$ arcs if and only if $q_{\\sigma}+s p_{\\sigma}\\equiv0$ (mod $n$ ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a ", "page_idx": 8}]
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{"preproc_blocks": [{"type": "text", "bbox": [109, 125, 499, 153], "lines": [{"bbox": [109, 127, 499, 142], "spans": [{"bbox": [109, 127, 306, 142], "score": 1.0, "content": "said to be canonical) on the cycles of ", "type": "text"}, {"bbox": [307, 130, 316, 138], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [317, 127, 414, 142], "score": 1.0, "content": " and on the graph ", "type": "text"}, {"bbox": [414, 129, 422, 138], "score": 0.87, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [422, 127, 499, 142], "score": 1.0, "content": ", by extending,", "type": "text"}], "index": 0}, {"bbox": [110, 143, 385, 155], "spans": [{"bbox": [110, 143, 129, 155], "score": 1.0, "content": "via ", "type": "text"}, {"bbox": [129, 147, 141, 155], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [141, 143, 241, 155], "score": 1.0, "content": ", the orientation of ", "type": "text"}, {"bbox": [241, 144, 255, 154], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [256, 143, 371, 155], "score": 1.0, "content": " to the other cycles of ", "type": "text"}, {"bbox": [371, 144, 381, 153], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 143, 385, 155], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 155, 500, 313], "lines": [{"bbox": [126, 155, 499, 171], "spans": [{"bbox": [126, 155, 332, 171], "score": 1.0, "content": "Property (i’) implies that the cycles of ", "type": "text"}, {"bbox": [332, 159, 342, 167], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [343, 155, 499, 171], "score": 1.0, "content": " naturally induce a cyclic per-", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 214, 186], "score": 1.0, "content": "mutation on the set ", "type": "text"}, {"bbox": [214, 172, 294, 185], "score": 0.95, "content": "\\mathcal{N}=\\{1,\\dotsc,d\\}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [294, 171, 501, 186], "score": 1.0, "content": " of the vertex labels. In fact, by walking", "type": "text"}], "index": 3}, {"bbox": [109, 185, 499, 199], "spans": [{"bbox": [109, 185, 492, 199], "score": 1.0, "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "type": "text"}, {"bbox": [493, 189, 499, 196], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}], "index": 4}, {"bbox": [109, 200, 501, 215], "spans": [{"bbox": [109, 200, 153, 215], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 202, 159, 213], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [159, 200, 285, 215], "score": 1.0, "content": ", one sequentially meets ", "type": "text"}, {"bbox": [285, 201, 291, 210], "score": 0.88, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [291, 200, 501, 215], "score": 1.0, "content": " vertices (whose labels are different from", "type": "text"}], "index": 5}, {"bbox": [110, 214, 501, 229], "spans": [{"bbox": [110, 214, 293, 229], "score": 1.0, "content": "each other), and then a new vertex ", "type": "text"}, {"bbox": [293, 216, 302, 225], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 214, 348, 229], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [349, 216, 354, 227], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [355, 214, 501, 229], "score": 1.0, "content": " which can be different from", "type": "text"}], "index": 6}, {"bbox": [110, 229, 500, 243], "spans": [{"bbox": [110, 232, 117, 240], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 229, 328, 243], "score": 1.0, "content": ". The sequence of the labellings of these ", "type": "text"}, {"bbox": [329, 231, 335, 240], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [335, 229, 500, 243], "score": 1.0, "content": " consecutive vertices defines the", "type": "text"}], "index": 7}, {"bbox": [110, 244, 499, 257], "spans": [{"bbox": [110, 244, 224, 257], "score": 1.0, "content": "cyclic permutation on ", "type": "text"}, {"bbox": [225, 245, 236, 254], "score": 0.89, "content": "\\mathcal{N}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [237, 244, 357, 257], "score": 1.0, "content": ". Further, each cycle of ", "type": "text"}, {"bbox": [358, 245, 368, 254], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [368, 244, 464, 257], "score": 1.0, "content": " precisely contains ", "type": "text"}, {"bbox": [464, 245, 499, 254], "score": 0.93, "content": "d^{\\prime}=l d", "type": "inline_equation", "height": 9, "width": 35}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 164, 272], "score": 1.0, "content": "arcs, with ", "type": "text"}, {"bbox": [164, 259, 190, 270], "score": 0.92, "content": "l\\geq1", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [190, 258, 219, 272], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [219, 259, 245, 268], "score": 0.93, "content": "l=1", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [245, 258, 500, 272], "score": 1.0, "content": " if and only if the 6-tuple satisfies (ii’) too. More-", "type": "text"}], "index": 9}, {"bbox": [109, 272, 500, 286], "spans": [{"bbox": [109, 272, 370, 286], "score": 1.0, "content": "over, property (i’) is independent from the integers ", "type": "text"}, {"bbox": [370, 277, 378, 282], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [378, 272, 403, 286], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [403, 277, 408, 282], "score": 0.86, "content": "s", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [409, 272, 500, 286], "score": 1.0, "content": "; hence, given two", "type": "text"}], "index": 10}, {"bbox": [109, 286, 502, 301], "spans": [{"bbox": [109, 286, 155, 301], "score": 1.0, "content": "6-tuples ", "type": "text"}, {"bbox": [155, 288, 250, 300], "score": 0.93, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [251, 286, 279, 301], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 288, 380, 300], "score": 0.93, "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [380, 286, 415, 301], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [415, 291, 423, 297], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [423, 286, 502, 301], "score": 1.0, "content": " satisfies (i’) if", "type": "text"}], "index": 11}, {"bbox": [110, 301, 246, 314], "spans": [{"bbox": [110, 301, 169, 314], "score": 1.0, "content": "and only if ", "type": "text"}, {"bbox": [169, 302, 179, 311], "score": 0.91, "content": "\\sigma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [180, 301, 246, 314], "score": 1.0, "content": " satisfies (i’).", "type": "text"}], "index": 12}], "index": 7}, {"type": "text", "bbox": [109, 313, 501, 471], "lines": [{"bbox": [125, 314, 498, 330], "spans": [{"bbox": [125, 314, 173, 330], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [174, 320, 181, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [181, 314, 417, 330], "score": 1.0, "content": " be a 6-tuple satisfying (i’) and suppose that ", "type": "text"}, {"bbox": [418, 317, 425, 326], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 314, 498, 330], "score": 1.0, "content": " is canonically", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 213, 344], "score": 1.0, "content": "oriented. An arc of ", "type": "text"}, {"bbox": [214, 331, 221, 340], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [222, 330, 293, 344], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [293, 331, 302, 340], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [303, 330, 500, 344], "score": 1.0, "content": " is said to be of type I if it is oriented", "type": "text"}], "index": 14}, {"bbox": [110, 345, 501, 358], "spans": [{"bbox": [110, 345, 190, 358], "score": 1.0, "content": "from a cycle of ", "type": "text"}, {"bbox": [190, 346, 200, 355], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [200, 345, 270, 358], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [271, 346, 283, 355], "score": 0.9, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [284, 345, 501, 358], "score": 1.0, "content": ", of type II if it is oriented from a cycle of", "type": "text"}], "index": 15}, {"bbox": [110, 358, 502, 374], "spans": [{"bbox": [110, 360, 122, 370], "score": 0.91, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [123, 358, 191, 374], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [192, 360, 202, 369], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [202, 358, 427, 374], "score": 1.0, "content": " and of type III otherwise (it joins cycles of ", "type": "text"}, {"bbox": [428, 360, 438, 369], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [438, 358, 502, 374], "score": 1.0, "content": " or cycles of", "type": "text"}], "index": 16}, {"bbox": [110, 372, 500, 387], "spans": [{"bbox": [110, 374, 127, 387], "score": 0.66, "content": "\\mathcal{C}^{\\prime\\prime})", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [127, 372, 500, 387], "score": 1.0, "content": ". Moreover, the arc is said to be of type I’ if it is oriented from a cycle", "type": "text"}], "index": 17}, {"bbox": [110, 387, 502, 406], "spans": [{"bbox": [110, 389, 122, 401], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [123, 387, 160, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [160, 389, 174, 401], "score": 0.88, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [174, 387, 235, 406], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [236, 389, 258, 402], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [259, 387, 296, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [296, 389, 319, 402], "score": 0.9, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [320, 387, 502, 406], "score": 1.0, "content": "), of type II’ if it is oriented from a", "type": "text"}], "index": 18}, {"bbox": [108, 401, 501, 419], "spans": [{"bbox": [108, 401, 138, 419], "score": 1.0, "content": "cycle ", "type": "text"}, {"bbox": [139, 403, 162, 416], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [162, 401, 198, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III’ otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii’).", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II’) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii’). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii’). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i’) and (ii’) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 153], "lines": [], "index": 0.5, "bbox_fs": [109, 127, 499, 155], "lines_deleted": true}, {"type": "text", "bbox": [109, 155, 500, 313], "lines": [{"bbox": [126, 155, 499, 171], "spans": [{"bbox": [126, 155, 332, 171], "score": 1.0, "content": "Property (i’) implies that the cycles of ", "type": "text"}, {"bbox": [332, 159, 342, 167], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [343, 155, 499, 171], "score": 1.0, "content": " naturally induce a cyclic per-", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 214, 186], "score": 1.0, "content": "mutation on the set ", "type": "text"}, {"bbox": [214, 172, 294, 185], "score": 0.95, "content": "\\mathcal{N}=\\{1,\\dotsc,d\\}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [294, 171, 501, 186], "score": 1.0, "content": " of the vertex labels. In fact, by walking", "type": "text"}], "index": 3}, {"bbox": [109, 185, 499, 199], "spans": [{"bbox": [109, 185, 492, 199], "score": 1.0, "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "type": "text"}, {"bbox": [493, 189, 499, 196], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}], "index": 4}, {"bbox": [109, 200, 501, 215], "spans": [{"bbox": [109, 200, 153, 215], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 202, 159, 213], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [159, 200, 285, 215], "score": 1.0, "content": ", one sequentially meets ", "type": "text"}, {"bbox": [285, 201, 291, 210], "score": 0.88, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [291, 200, 501, 215], "score": 1.0, "content": " vertices (whose labels are different from", "type": "text"}], "index": 5}, {"bbox": [110, 214, 501, 229], "spans": [{"bbox": [110, 214, 293, 229], "score": 1.0, "content": "each other), and then a new vertex ", "type": "text"}, {"bbox": [293, 216, 302, 225], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 214, 348, 229], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [349, 216, 354, 227], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [355, 214, 501, 229], "score": 1.0, "content": " which can be different from", "type": "text"}], "index": 6}, {"bbox": [110, 229, 500, 243], "spans": [{"bbox": [110, 232, 117, 240], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 229, 328, 243], "score": 1.0, "content": ". The sequence of the labellings of these ", "type": "text"}, {"bbox": [329, 231, 335, 240], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [335, 229, 500, 243], "score": 1.0, "content": " consecutive vertices defines the", "type": "text"}], "index": 7}, {"bbox": [110, 244, 499, 257], "spans": [{"bbox": [110, 244, 224, 257], "score": 1.0, "content": "cyclic permutation on ", "type": "text"}, {"bbox": [225, 245, 236, 254], "score": 0.89, "content": "\\mathcal{N}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [237, 244, 357, 257], "score": 1.0, "content": ". Further, each cycle of ", "type": "text"}, {"bbox": [358, 245, 368, 254], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [368, 244, 464, 257], "score": 1.0, "content": " precisely contains ", "type": "text"}, {"bbox": [464, 245, 499, 254], "score": 0.93, "content": "d^{\\prime}=l d", "type": "inline_equation", "height": 9, "width": 35}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 164, 272], "score": 1.0, "content": "arcs, with ", "type": "text"}, {"bbox": [164, 259, 190, 270], "score": 0.92, "content": "l\\geq1", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [190, 258, 219, 272], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [219, 259, 245, 268], "score": 0.93, "content": "l=1", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [245, 258, 500, 272], "score": 1.0, "content": " if and only if the 6-tuple satisfies (ii’) too. More-", "type": "text"}], "index": 9}, {"bbox": [109, 272, 500, 286], "spans": [{"bbox": [109, 272, 370, 286], "score": 1.0, "content": "over, property (i’) is independent from the integers ", "type": "text"}, {"bbox": [370, 277, 378, 282], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [378, 272, 403, 286], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [403, 277, 408, 282], "score": 0.86, "content": "s", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [409, 272, 500, 286], "score": 1.0, "content": "; hence, given two", "type": "text"}], "index": 10}, {"bbox": [109, 286, 502, 301], "spans": [{"bbox": [109, 286, 155, 301], "score": 1.0, "content": "6-tuples ", "type": "text"}, {"bbox": [155, 288, 250, 300], "score": 0.93, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [251, 286, 279, 301], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 288, 380, 300], "score": 0.93, "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [380, 286, 415, 301], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [415, 291, 423, 297], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [423, 286, 502, 301], "score": 1.0, "content": " satisfies (i’) if", "type": "text"}], "index": 11}, {"bbox": [110, 301, 246, 314], "spans": [{"bbox": [110, 301, 169, 314], "score": 1.0, "content": "and only if ", "type": "text"}, {"bbox": [169, 302, 179, 311], "score": 0.91, "content": "\\sigma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [180, 301, 246, 314], "score": 1.0, "content": " satisfies (i’).", "type": "text"}], "index": 12}], "index": 7, "bbox_fs": [109, 155, 502, 314]}, {"type": "text", "bbox": [109, 313, 501, 471], "lines": [{"bbox": [125, 314, 498, 330], "spans": [{"bbox": [125, 314, 173, 330], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [174, 320, 181, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [181, 314, 417, 330], "score": 1.0, "content": " be a 6-tuple satisfying (i’) and suppose that ", "type": "text"}, {"bbox": [418, 317, 425, 326], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 314, 498, 330], "score": 1.0, "content": " is canonically", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 213, 344], "score": 1.0, "content": "oriented. An arc of ", "type": "text"}, {"bbox": [214, 331, 221, 340], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [222, 330, 293, 344], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [293, 331, 302, 340], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [303, 330, 500, 344], "score": 1.0, "content": " is said to be of type I if it is oriented", "type": "text"}], "index": 14}, {"bbox": [110, 345, 501, 358], "spans": [{"bbox": [110, 345, 190, 358], "score": 1.0, "content": "from a cycle of ", "type": "text"}, {"bbox": [190, 346, 200, 355], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [200, 345, 270, 358], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [271, 346, 283, 355], "score": 0.9, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [284, 345, 501, 358], "score": 1.0, "content": ", of type II if it is oriented from a cycle of", "type": "text"}], "index": 15}, {"bbox": [110, 358, 502, 374], "spans": [{"bbox": [110, 360, 122, 370], "score": 0.91, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [123, 358, 191, 374], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [192, 360, 202, 369], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [202, 358, 427, 374], "score": 1.0, "content": " and of type III otherwise (it joins cycles of ", "type": "text"}, {"bbox": [428, 360, 438, 369], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [438, 358, 502, 374], "score": 1.0, "content": " or cycles of", "type": "text"}], "index": 16}, {"bbox": [110, 372, 500, 387], "spans": [{"bbox": [110, 374, 127, 387], "score": 0.66, "content": "\\mathcal{C}^{\\prime\\prime})", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [127, 372, 500, 387], "score": 1.0, "content": ". Moreover, the arc is said to be of type I’ if it is oriented from a cycle", "type": "text"}], "index": 17}, {"bbox": [110, 387, 502, 406], "spans": [{"bbox": [110, 389, 122, 401], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [123, 387, 160, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [160, 389, 174, 401], "score": 0.88, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [174, 387, 235, 406], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [236, 389, 258, 402], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [259, 387, 296, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [296, 389, 319, 402], "score": 0.9, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [320, 387, 502, 406], "score": 1.0, "content": "), of type II’ if it is oriented from a", "type": "text"}], "index": 18}, {"bbox": [108, 401, 501, 419], "spans": [{"bbox": [108, 401, 138, 419], "score": 1.0, "content": "cycle ", "type": "text"}, {"bbox": [139, 403, 162, 416], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [162, 401, 198, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III’ otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii’).", "type": "text"}], "index": 23}], "index": 18, "bbox_fs": [108, 314, 502, 473]}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II’) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5, "bbox_fs": [109, 473, 502, 561]}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii’). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii’). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i’) and (ii’) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5, "bbox_fs": [109, 560, 501, 676]}]}
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[{"type": "text", "bbox": [109, 125, 499, 153], "content": "", "index": 0}, {"type": "text", "bbox": [109, 155, 500, 313], "content": "Property (i’) implies that the cycles of naturally induce a cyclic per- mutation on the set of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex labelled , one sequentially meets vertices (whose labels are different from each other), and then a new vertex labelled which can be different from . The sequence of the labellings of these consecutive vertices defines the cyclic permutation on . Further, each cycle of precisely contains arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers and ; hence, given two 6-tuples and , then satisfies (i’) if and only if satisfies (i’).", "index": 1}, {"type": "text", "bbox": [109, 313, 501, 471], "content": "Let now be a 6-tuple satisfying (i’) and suppose that is canonically oriented. An arc of belonging to is said to be of type I if it is oriented from a cycle of to a cycle of , of type II if it is oriented from a cycle of to a cycle of and of type III otherwise (it joins cycles of or cycles of . Moreover, the arc is said to be of type I’ if it is oriented from a cycle (resp. ) to a cycle (resp. ), of type II’ if it is oriented from a cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it joins with ). Let be the set of the first arcs of , following the canonical orientation, starting from the arc coming out from the vertex of labelled . Obviously, the set contains all the arcs of if and only if the 6-tuple also satisfies (ii’).", "index": 2}, {"type": "text", "bbox": [110, 473, 500, 559], "content": "Now, denote by (resp. ) the number of the arcs of type I (resp. of type II) of and set . Similarly, denote by (resp. ) the number of the arcs of type (resp. of type II’) of and set . Note that has the same parity of and has the same parity of and hence of . It is evident that and only depend on the integers .", "index": 3}, {"type": "text", "bbox": [109, 559, 500, 675], "content": "The integers and give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle of , starting from a vertex and let be the cycle of containing . If is the first vertex with the same label of and if is the cycle of containing , we have . Thus, the cycle contains arcs if and only if (mod ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "index": 4}]
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[{"bbox": [126, 155, 499, 171], "content": "Property (i’) implies that the cycles of naturally induce a cyclic per-", "parent_index": 1, "line_index": 0}, {"bbox": [109, 171, 501, 186], "content": "mutation on the set of the vertex labels. In fact, by walking", "parent_index": 1, "line_index": 1}, {"bbox": [109, 185, 499, 199], "content": "along these canonically oriented cycles, starting from an arbitrary vertex", "parent_index": 1, "line_index": 2}, {"bbox": [109, 200, 501, 215], "content": "labelled , one sequentially meets vertices (whose labels are different from", "parent_index": 1, "line_index": 3}, {"bbox": [110, 214, 501, 229], "content": "each other), and then a new vertex labelled which can be different from", "parent_index": 1, "line_index": 4}, {"bbox": [110, 229, 500, 243], "content": ". The sequence of the labellings of these consecutive vertices defines the", "parent_index": 1, "line_index": 5}, {"bbox": [110, 244, 499, 257], "content": "cyclic permutation on . Further, each cycle of precisely contains", "parent_index": 1, "line_index": 6}, {"bbox": [110, 258, 500, 272], "content": "arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More-", "parent_index": 1, "line_index": 7}, {"bbox": [109, 272, 500, 286], "content": "over, property (i’) is independent from the integers and ; hence, given two", "parent_index": 1, "line_index": 8}, {"bbox": [109, 286, 502, 301], "content": "6-tuples and , then satisfies (i’) if", "parent_index": 1, "line_index": 9}, {"bbox": [110, 301, 246, 314], "content": "and only if satisfies (i’).", "parent_index": 1, "line_index": 10}, {"bbox": [125, 314, 498, 330], "content": "Let now be a 6-tuple satisfying (i’) and suppose that is canonically", "parent_index": 2, "line_index": 0}, {"bbox": [110, 330, 500, 344], "content": "oriented. An arc of belonging to is said to be of type I if it is oriented", "parent_index": 2, "line_index": 1}, {"bbox": [110, 345, 501, 358], "content": "from a cycle of to a cycle of , of type II if it is oriented from a cycle of", "parent_index": 2, "line_index": 2}, {"bbox": [110, 358, 502, 374], "content": "to a cycle of and of type III otherwise (it joins cycles of or cycles of", "parent_index": 2, "line_index": 3}, {"bbox": [110, 372, 500, 387], "content": ". Moreover, the arc is said to be of type I’ if it is oriented from a cycle", "parent_index": 2, "line_index": 4}, {"bbox": [110, 387, 502, 406], "content": "(resp. ) to a cycle (resp. ), of type II’ if it is oriented from a", "parent_index": 2, "line_index": 5}, {"bbox": [108, 401, 501, 419], "content": "cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it", "parent_index": 2, "line_index": 6}, {"bbox": [108, 417, 500, 430], "content": "joins with ). Let be the set of the first arcs of , following the", "parent_index": 2, "line_index": 7}, {"bbox": [108, 432, 502, 444], "content": "canonical orientation, starting from the arc coming out from the vertex of", "parent_index": 2, "line_index": 8}, {"bbox": [110, 446, 500, 459], "content": "labelled . Obviously, the set contains all the arcs of if and", "parent_index": 2, "line_index": 9}, {"bbox": [110, 461, 307, 473], "content": "only if the 6-tuple also satisfies (ii’).", "parent_index": 2, "line_index": 10}, {"bbox": [126, 473, 502, 490], "content": "Now, denote by (resp. ) the number of the arcs of type I (resp. of", "parent_index": 3, "line_index": 0}, {"bbox": [109, 489, 501, 503], "content": "type II) of and set . Similarly, denote by (resp. ) the", "parent_index": 3, "line_index": 1}, {"bbox": [109, 502, 500, 518], "content": "number of the arcs of type (resp. of type II’) of and set .", "parent_index": 3, "line_index": 2}, {"bbox": [109, 517, 499, 532], "content": "Note that has the same parity of and has the same parity of", "parent_index": 3, "line_index": 3}, {"bbox": [109, 532, 500, 546], "content": "and hence of . It is evident that and only depend on the integers", "parent_index": 3, "line_index": 4}, {"bbox": [110, 547, 154, 561], "content": ".", "parent_index": 3, "line_index": 5}, {"bbox": [127, 560, 500, 576], "content": "The integers and give an useful tool for verifying condition (ii’). In", "parent_index": 4, "line_index": 0}, {"bbox": [109, 575, 500, 590], "content": "fact, suppose to walk along the canonically oriented cycle of , starting", "parent_index": 4, "line_index": 1}, {"bbox": [110, 590, 500, 604], "content": "from a vertex and let be the cycle of containing . If is the first", "parent_index": 4, "line_index": 2}, {"bbox": [109, 604, 500, 618], "content": "vertex with the same label of and if is the cycle of containing ,", "parent_index": 4, "line_index": 3}, {"bbox": [109, 617, 501, 635], "content": "we have . Thus, the cycle contains arcs if and only", "parent_index": 4, "line_index": 4}, {"bbox": [109, 633, 500, 648], "content": "if (mod ). This proves that the 6-tuple satisfies (ii’). Thus,", "parent_index": 4, "line_index": 5}, {"bbox": [110, 648, 499, 662], "content": "(i’) and (ii’) are respectively, in a different language, conditions (i) and (ii)", "parent_index": 4, "line_index": 6}, {"bbox": [110, 662, 500, 676], "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "parent_index": 4, "line_index": 7}]
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[]
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6-tuple to be admissible when $d$ is odd. In fact, we have the following result:
Lemma 3 ([6], Theorem 2) Let $\sigma\,=\,(a,b,c,n,r,s)$ be a 6-tuple with $d\,=$ $2a+b+c$ odd. Then $\sigma$ is admissible if and only if it satisfies $(i\,?)$ and (ii’).
Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.
An immediate consequence of Lemma 3 is the following result:
Corollary 4 Let $\sigma=(a,b,c,n,r,s)$ be a $\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\sigma$ is admissible if and only if $\mathcal{D}$ has a unique cycle.
Proof. If $\sigma$ is admissible, then it is straightforward that $\mathcal{D}$ has a unique cycle. Vice versa, if $\mathcal{D}$ has a unique cycle, then (i’) holds. Since $n=1$ implies (ii’), the result is a direct consequence of the above lemma.
The parameter $p_{\sigma}$ associated to an admissible 6-tuple $\sigma$ is strictly related to the word $w(\sigma)$ associated to $\sigma$ . In fact, we have:
Lemma 5 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\,=\,w(\sigma)$ the associated word and $\varepsilon_{w}$ its exponent-sum. Then
$$
p_{\sigma}=\varepsilon_{w}.
$$
Proof. Since $\sigma$ is admissible, the arcs of $\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$ , with $u_{h}\,\in\,\{+1,-1\}$ . We have: $\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$ ,where $d+1=1$ . Since $u_{h}\!+\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows.
In [6] Dunwoody investigates a wide subclass of manifolds $M(\sigma)$ such that $p_{\sigma}=\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.
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<html><body>
<p data-bbox="109 125 499 140">6-tuple to be admissible when $d$ is odd. In fact, we have the following result: </p>
<p data-bbox="109 152 499 182">Lemma 3 ([6], Theorem 2) Let $\sigma\,=\,(a,b,c,n,r,s)$ be a 6-tuple with $d\,=$ $2a+b+c$ odd. Then $\sigma$ is admissible if and only if it satisfies $(i\,?)$ and (ii’). </p>
<p data-bbox="110 194 501 237">Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1. </p>
<p data-bbox="125 244 450 258">An immediate consequence of Lemma 3 is the following result: </p>
<p data-bbox="109 271 502 300">Corollary 4 Let $\sigma=(a,b,c,n,r,s)$ be a $\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\sigma$ is admissible if and only if $\mathcal{D}$ has a unique cycle. </p>
<p data-bbox="109 312 500 356">Proof. If $\sigma$ is admissible, then it is straightforward that $\mathcal{D}$ has a unique cycle. Vice versa, if $\mathcal{D}$ has a unique cycle, then (i’) holds. Since $n=1$ implies (ii’), the result is a direct consequence of the above lemma. </p>
<p data-bbox="109 357 501 385">The parameter $p_{\sigma}$ associated to an admissible 6-tuple $\sigma$ is strictly related to the word $w(\sigma)$ associated to $\sigma$ . In fact, we have: </p>
<p data-bbox="109 398 500 427">Lemma 5 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\,=\,w(\sigma)$ the associated word and $\varepsilon_{w}$ its exponent-sum. Then </p>
<div class="equation" data-bbox="282 445 326 454">$$
p_{\sigma}=\varepsilon_{w}.
$$</div>
<p data-bbox="109 466 501 555">Proof. Since $\sigma$ is admissible, the arcs of $\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$ , with $u_{h}\,\in\,\{+1,-1\}$ . We have: $\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$ ,where $d+1=1$ . Since $u_{h}\!+\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows. </p>
<p data-bbox="109 556 501 614">In [6] Dunwoody investigates a wide subclass of manifolds $M(\sigma)$ such that $p_{\sigma}=\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem. </p>
</body></html>
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0003042v1
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[{"type": "text", "text": "6-tuple to be admissible when $d$ is odd. In fact, we have the following result: ", "page_idx": 9}, {"type": "text", "text": "Lemma 3 ([6], Theorem 2) Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be a 6-tuple with $d\\,=$ $2a+b+c$ odd. Then $\\sigma$ is admissible if and only if it satisfies $(i\\,?)$ and (ii’). ", "page_idx": 9}, {"type": "text", "text": "Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1. ", "page_idx": 9}, {"type": "text", "text": "An immediate consequence of Lemma 3 is the following result: ", "page_idx": 9}, {"type": "text", "text": "Corollary 4 Let $\\sigma=(a,b,c,n,r,s)$ be a $\\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\\sigma$ is admissible if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 9}, {"type": "text", "text": "Proof. If $\\sigma$ is admissible, then it is straightforward that $\\mathcal{D}$ has a unique cycle. Vice versa, if $\\mathcal{D}$ has a unique cycle, then (i’) holds. Since $n=1$ implies (ii’), the result is a direct consequence of the above lemma. ", "page_idx": 9}, {"type": "text", "text": "The parameter $p_{\\sigma}$ associated to an admissible 6-tuple $\\sigma$ is strictly related to the word $w(\\sigma)$ associated to $\\sigma$ . In fact, we have: ", "page_idx": 9}, {"type": "text", "text": "Lemma 5 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\\,=\\,w(\\sigma)$ the associated word and $\\varepsilon_{w}$ its exponent-sum. Then ", "page_idx": 9}, {"type": "equation", "text": "$$\np_{\\sigma}=\\varepsilon_{w}.\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "Proof. Since $\\sigma$ is admissible, the arcs of $\\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}$ , with $u_{h}\\,\\in\\,\\{+1,-1\\}$ . We have: $\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}$ ,where $d+1=1$ . Since $u_{h}\\!+\\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows. ", "page_idx": 9}, {"type": "text", "text": "In [6] Dunwoody investigates a wide subclass of manifolds $M(\\sigma)$ such that $p_{\\sigma}=\\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem. ", "page_idx": 9}]
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{"preproc_blocks": [{"type": "text", "bbox": [109, 125, 499, 140], "lines": [{"bbox": [109, 127, 499, 141], "spans": [{"bbox": [109, 127, 265, 141], "score": 1.0, "content": "6-tuple to be admissible when ", "type": "text"}, {"bbox": [266, 129, 272, 138], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [272, 127, 499, 141], "score": 1.0, "content": " is odd. In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii’).", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i’) and (ii’), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i’) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii’), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [298, 691, 312, 702], "lines": [{"bbox": [297, 692, 312, 705], "spans": [{"bbox": [297, 692, 312, 705], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 140], "lines": [{"bbox": [109, 127, 499, 141], "spans": [{"bbox": [109, 127, 265, 141], "score": 1.0, "content": "6-tuple to be admissible when ", "type": "text"}, {"bbox": [266, 129, 272, 138], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [272, 127, 499, 141], "score": 1.0, "content": " is odd. In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0, "bbox_fs": [109, 127, 499, 141]}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii’).", "type": "text"}], "index": 2}], "index": 1.5, "bbox_fs": [108, 154, 501, 184]}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i’) and (ii’), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4, "bbox_fs": [110, 197, 499, 239]}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [127, 246, 449, 260]}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [110, 273, 501, 302]}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i’) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii’), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10, "bbox_fs": [109, 315, 500, 358]}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [110, 358, 500, 388]}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [109, 400, 501, 428]}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19, "bbox_fs": [105, 468, 504, 557]}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5, "bbox_fs": [109, 558, 500, 616]}]}
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[{"type": "text", "bbox": [109, 125, 499, 140], "content": "6-tuple to be admissible when is odd. In fact, we have the following result:", "index": 0}, {"type": "text", "bbox": [109, 152, 499, 182], "content": "Lemma 3 ([6], Theorem 2) Let be a 6-tuple with odd. Then is admissible if and only if it satisfies and (ii’).", "index": 1}, {"type": "text", "bbox": [110, 194, 501, 237], "content": "Remark 3. This result does not hold when is even. In fact, the 6-tuples , with even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.", "index": 2}, {"type": "text", "bbox": [125, 244, 450, 258], "content": "An immediate consequence of Lemma 3 is the following result:", "index": 3}, {"type": "text", "bbox": [109, 271, 502, 300], "content": "Corollary 4 Let be a -tuple with odd and . Then is admissible if and only if has a unique cycle.", "index": 4}, {"type": "text", "bbox": [109, 312, 500, 356], "content": "Proof. If is admissible, then it is straightforward that has a unique cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies (ii’), the result is a direct consequence of the above lemma.", "index": 5}, {"type": "text", "bbox": [109, 357, 501, 385], "content": "The parameter associated to an admissible 6-tuple is strictly related to the word associated to . In fact, we have:", "index": 6}, {"type": "text", "bbox": [109, 398, 500, 427], "content": "Lemma 5 Let be an admissible 6-tuple, the associated word and its exponent-sum. Then", "index": 7}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "content": "", "index": 8}, {"type": "text", "bbox": [109, 466, 501, 555], "content": "Proof. Since is admissible, the arcs of are precisely the arcs of orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have: ,where . Since if is of type I, if is of type II and if is of type III, the result immediately follows.", "index": 9}, {"type": "text", "bbox": [109, 556, 501, 614], "content": "In [6] Dunwoody investigates a wide subclass of manifolds such that and he conjectures that all the elements of this subclass are cyclic coverings of branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.", "index": 10}]
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[{"bbox": [109, 127, 499, 141], "content": "6-tuple to be admissible when is odd. In fact, we have the following result:", "parent_index": 0, "line_index": 0}, {"bbox": [108, 154, 501, 170], "content": "Lemma 3 ([6], Theorem 2) Let be a 6-tuple with", "parent_index": 1, "line_index": 0}, {"bbox": [110, 168, 497, 184], "content": "odd. Then is admissible if and only if it satisfies and (ii’).", "parent_index": 1, "line_index": 1}, {"bbox": [110, 197, 499, 210], "content": "Remark 3. This result does not hold when is even. In fact, the 6-tuples", "parent_index": 2, "line_index": 0}, {"bbox": [110, 212, 498, 225], "content": ", with even, satisfy (i’) and (ii’), but they are not admissible,", "parent_index": 2, "line_index": 1}, {"bbox": [110, 227, 255, 239], "content": "as pointed out in Remark 1.", "parent_index": 2, "line_index": 2}, {"bbox": [127, 246, 449, 260], "content": "An immediate consequence of Lemma 3 is the following result:", "parent_index": 3, "line_index": 0}, {"bbox": [110, 273, 501, 288], "content": "Corollary 4 Let be a -tuple with odd and", "parent_index": 4, "line_index": 0}, {"bbox": [110, 288, 439, 302], "content": ". Then is admissible if and only if has a unique cycle.", "parent_index": 4, "line_index": 1}, {"bbox": [126, 315, 500, 329], "content": "Proof. If is admissible, then it is straightforward that has a unique", "parent_index": 5, "line_index": 0}, {"bbox": [109, 330, 499, 344], "content": "cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies", "parent_index": 5, "line_index": 1}, {"bbox": [110, 344, 431, 358], "content": "(ii’), the result is a direct consequence of the above lemma.", "parent_index": 5, "line_index": 2}, {"bbox": [126, 358, 500, 375], "content": "The parameter associated to an admissible 6-tuple is strictly related", "parent_index": 6, "line_index": 0}, {"bbox": [110, 373, 374, 388], "content": "to the word associated to . In fact, we have:", "parent_index": 6, "line_index": 1}, {"bbox": [109, 400, 501, 415], "content": "Lemma 5 Let be an admissible 6-tuple, the", "parent_index": 7, "line_index": 0}, {"bbox": [110, 416, 355, 428], "content": "associated word and its exponent-sum. Then", "parent_index": 7, "line_index": 1}, {"bbox": [127, 468, 501, 481], "content": "Proof. Since is admissible, the arcs of are precisely the arcs of", "parent_index": 9, "line_index": 0}, {"bbox": [105, 482, 504, 518], "content": "orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have:", "parent_index": 9, "line_index": 1}, {"bbox": [110, 508, 499, 533], "content": ",where . Since", "parent_index": 9, "line_index": 2}, {"bbox": [109, 529, 500, 543], "content": "if is of type I, if is of type II and if is", "parent_index": 9, "line_index": 3}, {"bbox": [109, 543, 346, 557], "content": "of type III, the result immediately follows.", "parent_index": 9, "line_index": 4}, {"bbox": [126, 558, 500, 572], "content": "In [6] Dunwoody investigates a wide subclass of manifolds such that", "parent_index": 10, "line_index": 0}, {"bbox": [110, 573, 500, 586], "content": "and he conjectures that all the elements of this subclass are cyclic", "parent_index": 10, "line_index": 1}, {"bbox": [109, 586, 500, 601], "content": "coverings of branched over knots. In the next chapter this conjecture will", "parent_index": 10, "line_index": 2}, {"bbox": [109, 600, 375, 616], "content": "be proved as a corollary of a more general theorem.", "parent_index": 10, "line_index": 3}]
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[]
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[{"bbox": [266, 129, 272, 138], "content": "d", "parent_index": 0, "subtype": "inline"}, {"bbox": [284, 156, 380, 169], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [478, 156, 501, 167], "content": "d\\,=", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 171, 162, 181], "content": "2a+b+c", "parent_index": 1, "subtype": "inline"}, {"bbox": [222, 174, 229, 180], "content": "\\sigma", "parent_index": 1, "subtype": "inline"}, {"bbox": [429, 169, 447, 183], "content": "(i\\,?)", "parent_index": 1, "subtype": "inline"}, {"bbox": [342, 199, 349, 207], "content": "d", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 212, 181, 225], "content": "(1,0,c,1,2,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [213, 216, 218, 222], "content": "c", "parent_index": 2, "subtype": "inline"}, {"bbox": [202, 275, 294, 287], "content": "\\sigma=(a,b,c,n,r,s)", "parent_index": 4, "subtype": "inline"}, {"bbox": [322, 276, 329, 284], "content": "\\it6", "parent_index": 4, "subtype": "inline"}, {"bbox": [385, 275, 455, 285], "content": "d=2a+b+c", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 290, 139, 298], "content": "n=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [178, 293, 185, 298], "content": "\\sigma", "parent_index": 4, "subtype": "inline"}, {"bbox": [329, 290, 339, 298], "content": "\\mathcal{D}", "parent_index": 4, "subtype": "inline"}, {"bbox": [181, 320, 188, 326], "content": "\\sigma", "parent_index": 5, "subtype": "inline"}, {"bbox": [420, 317, 430, 326], "content": "\\mathcal{D}", "parent_index": 5, "subtype": "inline"}, {"bbox": [211, 332, 221, 340], "content": "\\mathcal{D}", "parent_index": 5, "subtype": "inline"}, {"bbox": [431, 332, 460, 340], "content": "n=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [207, 364, 218, 372], "content": "p_{\\sigma}", "parent_index": 6, "subtype": "inline"}, {"bbox": [402, 364, 409, 369], "content": "\\sigma", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 374, 199, 387], "content": "w(\\sigma)", "parent_index": 6, "subtype": "inline"}, {"bbox": [272, 378, 280, 384], "content": "\\sigma", "parent_index": 6, "subtype": "inline"}, {"bbox": [191, 402, 287, 414], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "parent_index": 7, "subtype": "inline"}, {"bbox": [426, 402, 479, 414], "content": "w\\,=\\,w(\\sigma)", "parent_index": 7, "subtype": "inline"}, {"bbox": [216, 420, 228, 427], "content": "\\varepsilon_{w}", "parent_index": 7, "subtype": "inline"}, {"bbox": [282, 445, 326, 454], "content": "p_{\\sigma}=\\varepsilon_{w}.", "parent_index": 8, "subtype": "interline"}, {"bbox": [206, 474, 213, 479], "content": "\\sigma", "parent_index": 9, "subtype": "inline"}, {"bbox": [356, 470, 366, 479], "content": "\\Delta", "parent_index": 9, "subtype": "inline"}, {"bbox": [110, 485, 125, 496], "content": "D_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [155, 488, 220, 496], "content": "e_{1},e_{2},\\ldots,e_{d}", "parent_index": 9, "subtype": "inline"}, {"bbox": [188, 500, 203, 510], "content": "D_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [252, 496, 327, 513], "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "parent_index": 9, "subtype": "inline"}, {"bbox": [362, 499, 440, 511], "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [110, 513, 302, 529], "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "parent_index": 9, "subtype": "inline"}, {"bbox": [338, 516, 383, 526], "content": "d+1=1", "parent_index": 9, "subtype": "inline"}, {"bbox": [421, 517, 499, 528], "content": "u_{h}\\!+\\!u_{h+1}=+2", "parent_index": 9, "subtype": "inline"}, {"bbox": [121, 534, 132, 541], "content": "e_{h}", "parent_index": 9, "subtype": "inline"}, {"bbox": [198, 531, 277, 542], "content": "u_{h}+u_{h+1}=-2", "parent_index": 9, "subtype": "inline"}, {"bbox": [291, 534, 302, 541], "content": "e_{h}", "parent_index": 9, "subtype": "inline"}, {"bbox": [392, 532, 462, 542], "content": "u_{h}+u_{h+1}=0", "parent_index": 9, "subtype": "inline"}, {"bbox": [476, 534, 487, 541], "content": "e_{h}", "parent_index": 9, "subtype": "inline"}, {"bbox": [420, 559, 449, 571], "content": "M(\\sigma)", "parent_index": 10, "subtype": "inline"}, {"bbox": [110, 574, 154, 585], "content": "p_{\\sigma}=\\pm1", "parent_index": 10, "subtype": "inline"}, {"bbox": [173, 587, 186, 597], "content": "\\mathbf{S^{3}}", "parent_index": 10, "subtype": "inline"}]
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[]
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# 3 Main results
The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots.
Theorem 6 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\,>\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\prime}$ is homeomorphic to:
i) $\mathbf{S^{3}}$ , if $p_{\sigma}=\pm1$ ,
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $p_{\sigma}=0$ ,
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|p_{\sigma}|$ , if $|p_{\sigma}|>1$ .
Proof. Since the two systems of curves $\mathcal{C}\,=\,\{C_{1},\ldots\,,C_{n}\}$ and $\mathcal{D}\,=$ $\{D_{1},...\,,D_{n}\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\prime}$ of genus $n$ , with $\partial U_{n}=\partial U_{n}^{\prime}=T_{n}$ , such that $M=U_{n}\cup U_{n}^{\prime}$ . Let now $\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\rho_{n}$ on $T_{n}$ . The action of $\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\prime}$ ) such that $\partial B_{1}\,=\,C_{1}$ (resp. $\partial B_{1}^{\prime}\;=\;D_{1}$ ). Since $\rho_{n}(C_{i})\,=\,C_{i+1}$ and $\rho_{n}(D_{i})\,=\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\,=\,\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\prime}\,=\,\rho_{n}^{k-1}(B_{1}^{\prime}))$ , for $k=1,\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\prime}$ ). By arguments contained in [38], the quotients $U_{1}\,=\,U_{n}/\mathcal{G}_{n}$ and $U_{1}^{\prime}\,=\,U_{n}^{\prime}/\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\prime}$ , having a genus one 1-bridge knot $K\,\subset\,M^{\prime}$ as singular set of order $n$ . Thus, $M^{\prime}$ is homeomorphic to $M/\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\prime}$ , branched over $K$ .
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<html><body>
<h1 data-bbox="109 121 247 140">3 Main results </h1>
<p data-bbox="110 151 500 209">The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots. </p>
<p data-bbox="110 222 500 294">Theorem 6 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\,>\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\prime}$ is homeomorphic to: </p>
<p data-bbox="126 295 384 339">i) $\mathbf{S^{3}}$ , if $p_{\sigma}=\pm1$ ,
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $p_{\sigma}=0$ ,
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|p_{\sigma}|$ , if $|p_{\sigma}|>1$ . </p>
<p data-bbox="109 350 501 669">Proof. Since the two systems of curves $\mathcal{C}\,=\,\{C_{1},\ldots\,,C_{n}\}$ and $\mathcal{D}\,=$ $\{D_{1},...\,,D_{n}\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\prime}$ of genus $n$ , with $\partial U_{n}=\partial U_{n}^{\prime}=T_{n}$ , such that $M=U_{n}\cup U_{n}^{\prime}$ . Let now $\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\rho_{n}$ on $T_{n}$ . The action of $\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\prime}$ ) such that $\partial B_{1}\,=\,C_{1}$ (resp. $\partial B_{1}^{\prime}\;=\;D_{1}$ ). Since $\rho_{n}(C_{i})\,=\,C_{i+1}$ and $\rho_{n}(D_{i})\,=\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\,=\,\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\prime}\,=\,\rho_{n}^{k-1}(B_{1}^{\prime}))$ , for $k=1,\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\prime}$ ). By arguments contained in [38], the quotients $U_{1}\,=\,U_{n}/\mathcal{G}_{n}$ and $U_{1}^{\prime}\,=\,U_{n}^{\prime}/\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\prime}$ , having a genus one 1-bridge knot $K\,\subset\,M^{\prime}$ as singular set of order $n$ . Thus, $M^{\prime}$ is homeomorphic to $M/\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\prime}$ , branched over $K$ . </p>
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0003042v1
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[{"type": "text", "text": "3 Main results ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots. ", "page_idx": 10}, {"type": "text", "text": "Theorem 6 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\\,>\\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\\prime}$ is homeomorphic to: ", "page_idx": 10}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $p_{\\sigma}=\\pm1$ , \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $p_{\\sigma}=0$ , \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=|p_{\\sigma}|$ , if $|p_{\\sigma}|>1$ . ", "page_idx": 10}, {"type": "text", "text": "Proof. Since the two systems of curves $\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}$ and $\\mathcal{D}\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\\prime}$ of genus $n$ , with $\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}$ , such that $M=U_{n}\\cup U_{n}^{\\prime}$ . Let now $\\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\\rho_{n}$ on $T_{n}$ . The action of $\\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\\prime}$ ) such that $\\partial B_{1}\\,=\\,C_{1}$ (resp. $\\partial B_{1}^{\\prime}\\;=\\;D_{1}$ ). Since $\\rho_{n}(C_{i})\\,=\\,C_{i+1}$ and $\\rho_{n}(D_{i})\\,=\\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))$ , for $k=1,\\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\\prime}$ ). By arguments contained in [38], the quotients $U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}$ and $U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\\prime}$ , having a genus one 1-bridge knot $K\\,\\subset\\,M^{\\prime}$ as singular set of order $n$ . Thus, $M^{\\prime}$ is homeomorphic to $M/\\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\\prime}$ , branched over $K$ . ", "page_idx": 10}]
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{"preproc_blocks": [{"type": "title", "bbox": [109, 121, 247, 140], "lines": [{"bbox": [110, 123, 246, 140], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [137, 123, 246, 140], "score": 1.0, "content": "Main results", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [110, 151, 500, 209], "lines": [{"bbox": [110, 154, 500, 168], "spans": [{"bbox": [110, 154, 500, 168], "score": 1.0, "content": "The following theorem is the main result of this paper and shows how the", "type": "text"}], "index": 1}, {"bbox": [110, 169, 500, 182], "spans": [{"bbox": [110, 169, 500, 182], "score": 1.0, "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "type": "text"}], "index": 2}, {"bbox": [109, 182, 499, 197], "spans": [{"bbox": [109, 182, 499, 197], "score": 1.0, "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "type": "text"}], "index": 3}, {"bbox": [109, 197, 381, 211], "spans": [{"bbox": [109, 197, 123, 211], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 198, 136, 208], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 197, 381, 211], "score": 1.0, "content": " or of lens spaces, branched over suitable knots.", "type": "text"}], "index": 4}], "index": 2.5}, {"type": "text", "bbox": [110, 222, 500, 294], "lines": [{"bbox": [109, 223, 500, 239], "spans": [{"bbox": [109, 223, 200, 239], "score": 1.0, "content": "Theorem 6 Let ", "type": "text"}, {"bbox": [201, 225, 297, 238], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [297, 223, 462, 239], "score": 1.0, "content": " be an admissible 6-tuple, with ", "type": "text"}, {"bbox": [462, 227, 496, 236], "score": 0.86, "content": "n\\,>\\,1", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [496, 223, 500, 239], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [111, 239, 501, 253], "spans": [{"bbox": [111, 239, 264, 253], "score": 1.0, "content": "Then the Dunwoody manifold ", "type": "text"}, {"bbox": [265, 239, 375, 252], "score": 0.92, "content": "M=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 110}, {"bbox": [376, 239, 411, 253], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [411, 240, 419, 250], "score": 0.46, "content": "n", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [420, 239, 501, 253], "score": 1.0, "content": "-fold cyclic cov-", "type": "text"}], "index": 6}, {"bbox": [110, 253, 501, 268], "spans": [{"bbox": [110, 253, 223, 268], "score": 1.0, "content": "ering of the manifold ", "type": "text"}, {"bbox": [224, 253, 338, 267], "score": 0.9, "content": "M^{\\prime}=M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [339, 253, 501, 268], "score": 1.0, "content": ", branched over a genus one 1-", "type": "text"}], "index": 7}, {"bbox": [111, 267, 499, 282], "spans": [{"bbox": [111, 268, 168, 282], "score": 1.0, "content": "bridge knot", "type": "text"}, {"bbox": [169, 267, 253, 281], "score": 0.93, "content": "K=K(a,b,c,r)", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [254, 268, 410, 282], "score": 1.0, "content": " only depending on the integers ", "type": "text"}, {"bbox": [411, 269, 449, 281], "score": 0.9, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [449, 268, 499, 282], "score": 1.0, "content": ". Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=|p_{\\sigma}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "|p_{\\sigma}|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". The intersection of these orbifolds is a 2-orbifold with two singular", "type": "text"}], "index": 26}, {"bbox": [109, 555, 499, 569], "spans": [{"bbox": [109, 555, 190, 569], "score": 1.0, "content": "points of order ", "type": "text"}, {"bbox": [190, 560, 197, 566], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [198, 555, 370, 569], "score": 1.0, "content": ", which is topologically the torus ", "type": "text"}, {"bbox": [370, 556, 430, 569], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [430, 555, 489, 569], "score": 1.0, "content": "; the curve ", "type": "text"}, {"bbox": [489, 557, 499, 566], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}], "index": 27}, {"bbox": [110, 570, 500, 584], "spans": [{"bbox": [110, 570, 145, 584], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". These curves, each of", "type": "text"}], "index": 29}, {"bbox": [109, 598, 500, 612], "spans": [{"bbox": [109, 598, 339, 612], "score": 1.0, "content": "which is a fundamental system of curves in ", "type": "text"}, {"bbox": [340, 600, 352, 611], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [352, 598, 500, 612], "score": 1.0, "content": ", define a Heegaard diagram", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 123, 627], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 614, 139, 624], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [139, 613, 206, 627], "score": 1.0, "content": " (induced by ", "type": "text"}, {"bbox": [207, 614, 289, 626], "score": 0.93, "content": "H(a,b,c,1,r,0))", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 613, 435, 627], "score": 1.0, "content": ". The union of the orbifolds ", "type": "text"}, {"bbox": [435, 615, 448, 625], "score": 0.93, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [448, 613, 474, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 614, 487, 627], "score": 0.93, "content": "U_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 613, 500, 627], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [110, 628, 500, 642], "spans": [{"bbox": [110, 628, 334, 642], "score": 1.0, "content": "a 3-orbifold topologically homeomorphic to ", "type": "text"}, {"bbox": [334, 629, 349, 638], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [350, 628, 500, 642], "score": 1.0, "content": ", having a genus one 1-bridge", "type": "text"}], "index": 32}, {"bbox": [109, 642, 501, 656], "spans": [{"bbox": [109, 642, 138, 656], "score": 1.0, "content": "knot ", "type": "text"}, {"bbox": [138, 644, 186, 653], "score": 0.93, "content": "K\\,\\subset\\,M^{\\prime}", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [186, 642, 319, 656], "score": 1.0, "content": " as singular set of order ", "type": "text"}, {"bbox": [319, 647, 326, 653], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [327, 642, 374, 656], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [374, 644, 389, 653], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [390, 642, 501, 656], "score": 1.0, "content": " is homeomorphic to", "type": "text"}], "index": 33}, {"bbox": [110, 656, 501, 671], "spans": [{"bbox": [110, 658, 141, 670], "score": 0.94, "content": "M/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [141, 656, 203, 671], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [203, 658, 216, 667], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [216, 656, 254, 671], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [254, 662, 261, 667], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [262, 656, 383, 671], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [383, 658, 399, 667], "score": 0.91, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [399, 656, 484, 671], "score": 1.0, "content": ", branched over ", "type": "text"}, {"bbox": [484, 659, 496, 667], "score": 0.91, "content": "K", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [496, 656, 501, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 23.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 310, 702], "lines": [{"bbox": [298, 692, 312, 704], "spans": [{"bbox": [298, 692, 312, 704], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 247, 140], "lines": [{"bbox": [110, 123, 246, 140], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [137, 123, 246, 140], "score": 1.0, "content": "Main results", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [110, 151, 500, 209], "lines": [{"bbox": [110, 154, 500, 168], "spans": [{"bbox": [110, 154, 500, 168], "score": 1.0, "content": "The following theorem is the main result of this paper and shows how the", "type": "text"}], "index": 1}, {"bbox": [110, 169, 500, 182], "spans": [{"bbox": [110, 169, 500, 182], "score": 1.0, "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "type": "text"}], "index": 2}, {"bbox": [109, 182, 499, 197], "spans": [{"bbox": [109, 182, 499, 197], "score": 1.0, "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "type": "text"}], "index": 3}, {"bbox": [109, 197, 381, 211], "spans": [{"bbox": [109, 197, 123, 211], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 198, 136, 208], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 197, 381, 211], "score": 1.0, "content": " or of lens spaces, branched over suitable knots.", "type": "text"}], "index": 4}], "index": 2.5, "bbox_fs": [109, 154, 500, 211]}, {"type": "text", "bbox": [110, 222, 500, 294], "lines": [{"bbox": [109, 223, 500, 239], "spans": [{"bbox": [109, 223, 200, 239], "score": 1.0, "content": "Theorem 6 Let ", "type": "text"}, {"bbox": [201, 225, 297, 238], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [297, 223, 462, 239], "score": 1.0, "content": " be an admissible 6-tuple, with ", "type": "text"}, {"bbox": [462, 227, 496, 236], "score": 0.86, "content": "n\\,>\\,1", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [496, 223, 500, 239], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [111, 239, 501, 253], "spans": [{"bbox": [111, 239, 264, 253], "score": 1.0, "content": "Then the Dunwoody manifold ", "type": "text"}, {"bbox": [265, 239, 375, 252], "score": 0.92, "content": "M=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 110}, {"bbox": [376, 239, 411, 253], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [411, 240, 419, 250], "score": 0.46, "content": "n", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [420, 239, 501, 253], "score": 1.0, "content": "-fold cyclic cov-", "type": "text"}], "index": 6}, {"bbox": [110, 253, 501, 268], "spans": [{"bbox": [110, 253, 223, 268], "score": 1.0, "content": "ering of the manifold ", "type": "text"}, {"bbox": [224, 253, 338, 267], "score": 0.9, "content": "M^{\\prime}=M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [339, 253, 501, 268], "score": 1.0, "content": ", branched over a genus one 1-", "type": "text"}], "index": 7}, {"bbox": [111, 267, 499, 282], "spans": [{"bbox": [111, 268, 168, 282], "score": 1.0, "content": "bridge knot", "type": "text"}, {"bbox": [169, 267, 253, 281], "score": 0.93, "content": "K=K(a,b,c,r)", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [254, 268, 410, 282], "score": 1.0, "content": " only depending on the integers ", "type": "text"}, {"bbox": [411, 269, 449, 281], "score": 0.9, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [449, 268, 499, 282], "score": 1.0, "content": ". Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7, "bbox_fs": [109, 223, 501, 296]}, {"type": "list", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10, "is_list_end_line": true}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=|p_{\\sigma}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "|p_{\\sigma}|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12, "is_list_start_line": true}], "index": 11, "bbox_fs": [127, 296, 382, 339]}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". The intersection of these orbifolds is a 2-orbifold with two singular", "type": "text"}], "index": 26}, {"bbox": [109, 555, 499, 569], "spans": [{"bbox": [109, 555, 190, 569], "score": 1.0, "content": "points of order ", "type": "text"}, {"bbox": [190, 560, 197, 566], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [198, 555, 370, 569], "score": 1.0, "content": ", which is topologically the torus ", "type": "text"}, {"bbox": [370, 556, 430, 569], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [430, 555, 489, 569], "score": 1.0, "content": "; the curve ", "type": "text"}, {"bbox": [489, 557, 499, 566], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}], "index": 27}, {"bbox": [110, 570, 500, 584], "spans": [{"bbox": [110, 570, 145, 584], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". These curves, each of", "type": "text"}], "index": 29}, {"bbox": [109, 598, 500, 612], "spans": [{"bbox": [109, 598, 339, 612], "score": 1.0, "content": "which is a fundamental system of curves in ", "type": "text"}, {"bbox": [340, 600, 352, 611], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [352, 598, 500, 612], "score": 1.0, "content": ", define a Heegaard diagram", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 123, 627], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 614, 139, 624], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [139, 613, 206, 627], "score": 1.0, "content": " (induced by ", "type": "text"}, {"bbox": [207, 614, 289, 626], "score": 0.93, "content": "H(a,b,c,1,r,0))", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 613, 435, 627], "score": 1.0, "content": ". The union of the orbifolds ", "type": "text"}, {"bbox": [435, 615, 448, 625], "score": 0.93, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [448, 613, 474, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 614, 487, 627], "score": 0.93, "content": "U_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 613, 500, 627], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [110, 628, 500, 642], "spans": [{"bbox": [110, 628, 334, 642], "score": 1.0, "content": "a 3-orbifold topologically homeomorphic to ", "type": "text"}, {"bbox": [334, 629, 349, 638], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [350, 628, 500, 642], "score": 1.0, "content": ", having a genus one 1-bridge", "type": "text"}], "index": 32}, {"bbox": [109, 642, 501, 656], "spans": [{"bbox": [109, 642, 138, 656], "score": 1.0, "content": "knot ", "type": "text"}, {"bbox": [138, 644, 186, 653], "score": 0.93, "content": "K\\,\\subset\\,M^{\\prime}", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [186, 642, 319, 656], "score": 1.0, "content": " as singular set of order ", "type": "text"}, {"bbox": [319, 647, 326, 653], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [327, 642, 374, 656], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [374, 644, 389, 653], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [390, 642, 501, 656], "score": 1.0, "content": " is homeomorphic to", "type": "text"}], "index": 33}, {"bbox": [110, 656, 501, 671], "spans": [{"bbox": [110, 658, 141, 670], "score": 0.94, "content": "M/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [141, 656, 203, 671], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [203, 658, 216, 667], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [216, 656, 254, 671], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [254, 662, 261, 667], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [262, 656, 383, 671], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [383, 658, 399, 667], "score": 0.91, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [399, 656, 484, 671], "score": 1.0, "content": ", branched over ", "type": "text"}, {"bbox": [484, 659, 496, 667], "score": 0.91, "content": "K", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [496, 656, 501, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 23.5, "bbox_fs": [108, 352, 503, 671]}]}
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[{"type": "title", "bbox": [109, 121, 247, 140], "content": "3 Main results", "index": 0}, {"type": "text", "bbox": [110, 151, 500, 209], "content": "The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of or of lens spaces, branched over suitable knots.", "index": 1}, {"type": "text", "bbox": [110, 222, 500, 294], "content": "Theorem 6 Let be an admissible 6-tuple, with . Then the Dunwoody manifold is the -fold cyclic cov- ering of the manifold , branched over a genus one 1- bridge knot only depending on the integers . Further, is homeomorphic to:", "index": 2}, {"type": "list", "bbox": [126, 295, 384, 339], "content": "", "index": 3}, {"type": "text", "bbox": [109, 350, 501, 669], "content": "Proof. Since the two systems of curves and on define a Heegaard diagram of , there exist two handle- bodies and of genus , with , such that . Let now be the cyclic group of order generated by the homeomorphism on . The action of on extends to both the handlebodies and (see [29]), and hence to the 3-manifold . Let (resp. ) be a disc properly embedded in (resp. in ) such that (resp. ). Since and (mod ), the discs (resp. , for , form a system of meridian discs for the handlebody (resp. ). By arguments contained in [38], the quotients and are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der . The intersection of these orbifolds is a 2-orbifold with two singular points of order , which is topologically the torus ; the curve (resp. ), which is the image via the quotient map of the curves (resp. of the curves ), is non-homotopically trivial in . These curves, each of which is a fundamental system of curves in , define a Heegaard diagram of (induced by . The union of the orbifolds and is a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge knot as singular set of order . Thus, is homeomorphic to and hence is the -fold cyclic covering of , branched over .", "index": 4}]
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[{"bbox": [110, 123, 246, 140], "content": "3 Main results", "parent_index": 0, "line_index": 0}, {"bbox": [110, 154, 500, 168], "content": "The following theorem is the main result of this paper and shows how the", "parent_index": 1, "line_index": 0}, {"bbox": [110, 169, 500, 182], "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "parent_index": 1, "line_index": 1}, {"bbox": [109, 182, 499, 197], "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "parent_index": 1, "line_index": 2}, {"bbox": [109, 197, 381, 211], "content": "of or of lens spaces, branched over suitable knots.", "parent_index": 1, "line_index": 3}, {"bbox": [109, 223, 500, 239], "content": "Theorem 6 Let be an admissible 6-tuple, with .", "parent_index": 2, "line_index": 0}, {"bbox": [111, 239, 501, 253], "content": "Then the Dunwoody manifold is the -fold cyclic cov-", "parent_index": 2, "line_index": 1}, {"bbox": [110, 253, 501, 268], "content": "ering of the manifold , branched over a genus one 1-", "parent_index": 2, "line_index": 2}, {"bbox": [111, 267, 499, 282], "content": "bridge knot only depending on the integers . Further,", "parent_index": 2, "line_index": 3}, {"bbox": [110, 282, 235, 296], "content": "is homeomorphic to:", "parent_index": 2, "line_index": 4}, {"bbox": [127, 296, 218, 311], "content": "i) , if ,", "parent_index": 3, "line_index": 0}, {"bbox": [127, 310, 241, 325], "content": "ii) , if ,", "parent_index": 3, "line_index": 1}, {"bbox": [127, 325, 382, 339], "content": "iii) a lens space with , if .", "parent_index": 3, "line_index": 2}, {"bbox": [126, 352, 500, 368], "content": "Proof. Since the two systems of curves and", "parent_index": 4, "line_index": 0}, {"bbox": [110, 367, 498, 382], "content": "on define a Heegaard diagram of , there exist two handle-", "parent_index": 4, "line_index": 1}, {"bbox": [108, 380, 498, 397], "content": "bodies and of genus , with , such that .", "parent_index": 4, "line_index": 2}, {"bbox": [109, 397, 500, 411], "content": "Let now be the cyclic group of order generated by the homeomorphism", "parent_index": 4, "line_index": 3}, {"bbox": [110, 410, 498, 426], "content": "on . The action of on extends to both the handlebodies", "parent_index": 4, "line_index": 4}, {"bbox": [110, 426, 500, 439], "content": "and (see [29]), and hence to the 3-manifold . Let (resp. ) be", "parent_index": 4, "line_index": 5}, {"bbox": [108, 439, 500, 455], "content": "a disc properly embedded in (resp. in ) such that (resp.", "parent_index": 4, "line_index": 6}, {"bbox": [110, 453, 501, 470], "content": "). Since and (mod ), the discs", "parent_index": 4, "line_index": 7}, {"bbox": [110, 466, 503, 485], "content": "(resp. , for , form a system of", "parent_index": 4, "line_index": 8}, {"bbox": [108, 483, 501, 498], "content": "meridian discs for the handlebody (resp. ). By arguments contained", "parent_index": 4, "line_index": 9}, {"bbox": [108, 497, 500, 512], "content": "in [38], the quotients and are both handlebody", "parent_index": 4, "line_index": 10}, {"bbox": [110, 513, 501, 527], "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "parent_index": 4, "line_index": 11}, {"bbox": [108, 526, 501, 542], "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "parent_index": 4, "line_index": 12}, {"bbox": [110, 542, 500, 554], "content": "der . The intersection of these orbifolds is a 2-orbifold with two singular", "parent_index": 4, "line_index": 13}, {"bbox": [109, 555, 499, 569], "content": "points of order , which is topologically the torus ; the curve", "parent_index": 4, "line_index": 14}, {"bbox": [110, 570, 500, 584], "content": "(resp. ), which is the image via the quotient map of the curves (resp.", "parent_index": 4, "line_index": 15}, {"bbox": [109, 584, 502, 598], "content": "of the curves ), is non-homotopically trivial in . These curves, each of", "parent_index": 4, "line_index": 16}, {"bbox": [109, 598, 500, 612], "content": "which is a fundamental system of curves in , define a Heegaard diagram", "parent_index": 4, "line_index": 17}, {"bbox": [110, 613, 500, 627], "content": "of (induced by . The union of the orbifolds and is", "parent_index": 4, "line_index": 18}, {"bbox": [110, 628, 500, 642], "content": "a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge", "parent_index": 4, "line_index": 19}, {"bbox": [109, 642, 501, 656], "content": "knot as singular set of order . Thus, is homeomorphic to", "parent_index": 4, "line_index": 20}, {"bbox": [110, 656, 501, 671], "content": "and hence is the -fold cyclic covering of , branched over .", "parent_index": 4, "line_index": 21}]
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[{"bbox": [123, 198, 136, 208], "content": "\\mathrm{{S^{3}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [201, 225, 297, 238], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "parent_index": 2, "subtype": "inline"}, {"bbox": [462, 227, 496, 236], "content": "n\\,>\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [265, 239, 375, 252], "content": "M=M(a,b,c,n,r,s)", "parent_index": 2, "subtype": "inline"}, {"bbox": [411, 240, 419, 250], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [224, 253, 338, 267], "content": "M^{\\prime}=M(a,b,c,1,r,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [169, 267, 253, 281], "content": "K=K(a,b,c,r)", "parent_index": 2, "subtype": "inline"}, {"bbox": [411, 269, 449, 281], "content": "a,b,c,r", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 284, 126, 293], "content": "M^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [139, 297, 154, 308], "content": "\\mathbf{S^{3}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [172, 297, 215, 310], "content": "p_{\\sigma}=\\pm1", "parent_index": 3, "subtype": "inline"}, {"bbox": [144, 311, 185, 323], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [203, 311, 237, 324], "content": "p_{\\sigma}=0", "parent_index": 3, "subtype": "inline"}, {"bbox": [212, 325, 249, 339], "content": "L(\\alpha,\\beta)", "parent_index": 3, "subtype": "inline"}, {"bbox": [278, 325, 321, 339], "content": "\\alpha=|p_{\\sigma}|", "parent_index": 3, "subtype": "inline"}, {"bbox": [339, 325, 379, 339], "content": "|p_{\\sigma}|>1", "parent_index": 3, "subtype": "inline"}, {"bbox": [348, 354, 445, 367], "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "parent_index": 4, "subtype": "inline"}, {"bbox": [474, 354, 500, 366], "content": "\\mathcal{D}\\,=", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 369, 180, 381], "content": "\\{D_{1},...\\,,D_{n}\\}", "parent_index": 4, "subtype": "inline"}, {"bbox": [199, 369, 212, 380], "content": "T_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [365, 369, 377, 378], "content": "M", "parent_index": 4, "subtype": "inline"}, {"bbox": [146, 384, 160, 394], "content": "U_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 383, 199, 396], "content": "U_{n}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [246, 385, 254, 393], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [286, 382, 372, 396], "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [429, 383, 496, 396], "content": "M=U_{n}\\cup U_{n}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [155, 398, 168, 409], "content": "\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [315, 401, 322, 407], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 416, 122, 424], "content": "\\rho_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 413, 158, 423], "content": "T_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [248, 413, 261, 423], "content": "\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [284, 413, 297, 423], "content": "T_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [485, 413, 498, 423], "content": "U_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [134, 427, 147, 439], "content": "U_{n}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [362, 427, 374, 436], "content": "M", "parent_index": 4, "subtype": "inline"}, {"bbox": [408, 427, 421, 438], "content": "B_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [464, 427, 477, 439], "content": "B_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [264, 442, 278, 452], "content": "U_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [335, 441, 349, 453], "content": "U_{n}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [412, 442, 465, 452], "content": "\\partial B_{1}\\,=\\,C_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 456, 165, 468], "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [214, 455, 289, 468], "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [318, 455, 396, 468], "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [433, 459, 440, 465], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 469, 187, 482], "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 469, 308, 482], "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "parent_index": 4, "subtype": "inline"}, {"bbox": [336, 470, 403, 482], "content": "k=1,\\dotsc,n", "parent_index": 4, "subtype": "inline"}, {"bbox": [291, 485, 305, 495], "content": "U_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [345, 484, 359, 496], "content": "U_{n}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [225, 498, 290, 511], "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [319, 498, 384, 511], "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [131, 546, 138, 551], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [190, 560, 197, 566], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [370, 556, 430, 569], "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [489, 557, 499, 566], "content": "C", "parent_index": 4, "subtype": "inline"}, {"bbox": [146, 572, 156, 581], "content": "D", "parent_index": 4, "subtype": "inline"}, {"bbox": [454, 572, 466, 582], "content": "C_{i}", "parent_index": 4, "subtype": "inline"}, {"bbox": [181, 586, 194, 597], "content": "D_{i}", "parent_index": 4, "subtype": "inline"}, {"bbox": [367, 586, 379, 596], "content": "T_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [340, 600, 352, 611], "content": "T_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 614, 139, 624], "content": "M^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [207, 614, 289, 626], "content": "H(a,b,c,1,r,0))", "parent_index": 4, "subtype": "inline"}, {"bbox": [435, 615, 448, 625], "content": "U_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [474, 614, 487, 627], "content": "U_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [334, 629, 349, 638], "content": "M^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [138, 644, 186, 653], "content": "K\\,\\subset\\,M^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [319, 647, 326, 653], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [374, 644, 389, 653], "content": "M^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 658, 141, 670], "content": "M/\\mathcal{G}_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [203, 658, 216, 667], "content": "M", "parent_index": 4, "subtype": "inline"}, {"bbox": [254, 662, 261, 667], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [383, 658, 399, 667], "content": "M^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [484, 659, 496, 667], "content": "K", "parent_index": 4, "subtype": "inline"}]
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Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\prime}$ follows from Proposition 2 and Lemma 5.
Remark 4. More generally, given two positive integers $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\prime}$ -fold cyclic covering of the manifold $M^{\prime}=$ $M(a,b,c,n^{\prime},r,s)$ , branched over an $(n^{\prime},1)$ -knot in $M^{\prime}$ .
Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively.
Let us consider now the class of the Dunwoody manifolds $\textstyle M_{n}\ =$ $M(a,b,c,n,r,s)$ with $p=\pm1$ (and hence $d$ odd) and $s\,=\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.
Corollary 7 Let $\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\sigma_{1}}=\pm1$ and $s=-p_{\sigma_{1}}q_{\sigma_{1}}$ . Then the $\it6$ -tuple $\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\mathrm{{S^{3}}}$ , branched over a genus one $\mathit{1}$ -bridge knot $K\subset{\bf S^{3}}$ , which is independent on $n$ .
Proof. Obviously $(a,b,c,1,r,s)=\sigma_{1}$ . Since $\sigma_{1}$ is admissible, it satisfies (i’). This proves that $\sigma_{n}$ satisfies $\left(\mathrm{i}^{\,\circ}\right)$ , for each $n>1$ . Since $s=-p_{\sigma_{1}}q_{\sigma_{1}}=$ $-p_{\sigma_{n}}q_{\sigma_{n}}$ and $p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$ , we obtain $q_{\sigma_{n}}+s p_{\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6.
We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].
An interesting problem which naturally arises is that of characterizing the set $\kappa$ of branching knots in $\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\alpha$ and $\beta$ , with $\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\alpha,\beta)$ is equivalent to the 2-bridge knot of type $(\alpha,\alpha-\beta)$ , then $\beta$ can be assumed to be even.
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<p data-bbox="110 125 500 168">Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\prime}$ follows from Proposition 2 and Lemma 5. </p>
<p data-bbox="110 173 500 232">Remark 4. More generally, given two positive integers $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\prime}$ -fold cyclic covering of the manifold $M^{\prime}=$ $M(a,b,c,n^{\prime},r,s)$ , branched over an $(n^{\prime},1)$ -knot in $M^{\prime}$ . </p>
<p data-bbox="109 237 500 309">Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively. </p>
<p data-bbox="110 315 500 387">Let us consider now the class of the Dunwoody manifolds $\textstyle M_{n}\ =$ $M(a,b,c,n,r,s)$ with $p=\pm1$ (and hence $d$ odd) and $s\,=\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture. </p>
<p data-bbox="110 398 500 470">Corollary 7 Let $\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\sigma_{1}}=\pm1$ and $s=-p_{\sigma_{1}}q_{\sigma_{1}}$ . Then the $\it6$ -tuple $\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\mathrm{{S^{3}}}$ , branched over a genus one $\mathit{1}$ -bridge knot $K\subset{\bf S^{3}}$ , which is independent on $n$ . </p>
<p data-bbox="109 481 500 582">Proof. Obviously $(a,b,c,1,r,s)=\sigma_{1}$ . Since $\sigma_{1}$ is admissible, it satisfies (i’). This proves that $\sigma_{n}$ satisfies $\left(\mathrm{i}^{\,\circ}\right)$ , for each $n>1$ . Since $s=-p_{\sigma_{1}}q_{\sigma_{1}}=$ $-p_{\sigma_{n}}q_{\sigma_{n}}$ and $p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$ , we obtain $q_{\sigma_{n}}+s p_{\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6. </p>
<p data-bbox="109 587 500 617">We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32]. </p>
<p data-bbox="109 618 500 674">An interesting problem which naturally arises is that of characterizing the set $\kappa$ of branching knots in $\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\alpha$ and $\beta$ , with $\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\alpha,\beta)$ is equivalent to the 2-bridge knot of type $(\alpha,\alpha-\beta)$ , then $\beta$ can be assumed to be even. </p>
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0003042v1
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[{"type": "text", "text": "Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\\prime}$ follows from Proposition 2 and Lemma 5. ", "page_idx": 11}, {"type": "text", "text": "Remark 4. More generally, given two positive integers $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\\prime}$ -fold cyclic covering of the manifold $M^{\\prime}=$ $M(a,b,c,n^{\\prime},r,s)$ , branched over an $(n^{\\prime},1)$ -knot in $M^{\\prime}$ . ", "page_idx": 11}, {"type": "text", "text": "Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively. ", "page_idx": 11}, {"type": "text", "text": "Let us consider now the class of the Dunwoody manifolds $\\textstyle M_{n}\\ =$ $M(a,b,c,n,r,s)$ with $p=\\pm1$ (and hence $d$ odd) and $s\\,=\\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture. ", "page_idx": 11}, {"type": "text", "text": "Corollary 7 Let $\\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\\sigma_{1}}=\\pm1$ and $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}$ . Then the $\\it6$ -tuple $\\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over a genus one $\\mathit{1}$ -bridge knot $K\\subset{\\bf S^{3}}$ , which is independent on $n$ . ", "page_idx": 11}, {"type": "text", "text": "Proof. Obviously $(a,b,c,1,r,s)=\\sigma_{1}$ . Since $\\sigma_{1}$ is admissible, it satisfies (i’). This proves that $\\sigma_{n}$ satisfies $\\left(\\mathrm{i}^{\\,\\circ}\\right)$ , for each $n>1$ . Since $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=$ $-p_{\\sigma_{n}}q_{\\sigma_{n}}$ and $p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1$ , we obtain $q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6. ", "page_idx": 11}, {"type": "text", "text": "We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32]. ", "page_idx": 11}, {"type": "text", "text": "An interesting problem which naturally arises is that of characterizing the set $\\kappa$ of branching knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\\alpha$ and $\\beta$ , with $\\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\\alpha,\\beta)$ is equivalent to the 2-bridge knot of type $(\\alpha,\\alpha-\\beta)$ , then $\\beta$ can be assumed to be even. ", "page_idx": 11}]
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The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i’). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}], "index": 32.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 438, 142], "score": 1.0, "content": "Since the handlebody orbifolds and their gluing only depend on ", "type": "text"}, {"bbox": [438, 129, 476, 141], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [476, 128, 500, 142], "score": 1.0, "content": ", the", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 277, 156], "score": 1.0, "content": "same holds for the branching set ", "type": "text"}, {"bbox": [277, 144, 288, 153], "score": 0.9, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [289, 142, 445, 156], "score": 1.0, "content": ". The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [109, 128, 500, 169]}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "bbox_fs": [109, 176, 500, 234]}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9, "bbox_fs": [109, 239, 500, 312]}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14, "bbox_fs": [109, 316, 501, 389]}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19, "bbox_fs": [109, 399, 502, 473]}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i’). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25, "bbox_fs": [109, 483, 501, 583]}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5, "bbox_fs": [110, 589, 499, 618]}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 141], "spans": [{"bbox": [110, 128, 500, 141], "score": 1.0, "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [111, 142, 500, 156], "spans": [{"bbox": [111, 142, 297, 156], "score": 1.0, "content": "[30]. Since the 2-bridge knot of type ", "type": "text", "cross_page": true}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28, "cross_page": true}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text", "cross_page": true}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51, "cross_page": true}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text", "cross_page": true}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8, "cross_page": true}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text", "cross_page": true}], "index": 2}], "index": 32.5, "bbox_fs": [109, 618, 501, 676]}]}
|
[{"type": "text", "bbox": [110, 125, 500, 168], "content": "Since the handlebody orbifolds and their gluing only depend on , the same holds for the branching set . The homeomorphism type of follows from Proposition 2 and Lemma 5.", "index": 0}, {"type": "text", "bbox": [110, 173, 500, 232], "content": "Remark 4. More generally, given two positive integers and such that divides , if is admissible, then the Dunwoody man- ifold is the -fold cyclic covering of the manifold , branched over an -knot in .", "index": 1}, {"type": "text", "bbox": [109, 237, 500, 309], "content": "Example 2. The Dunwoody manifolds , and , with coprime, are -fold cyclic coverings of the manifolds , and respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of copies of , and respectively.", "index": 2}, {"type": "text", "bbox": [110, 315, 500, 387], "content": "Let us consider now the class of the Dunwoody manifolds with (and hence odd) and . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are -fold cyclic coverings of , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.", "index": 3}, {"type": "text", "bbox": [110, 398, 500, 470], "content": "Corollary 7 Let be an admissible 6-tuple with and . Then the -tuple is admissible for each and the Dunwoody manifold is a n-fold cyclic coverings of , branched over a genus one -bridge knot , which is independent on .", "index": 4}, {"type": "text", "bbox": [109, 481, 500, 582], "content": "Proof. Obviously . Since is admissible, it satisfies (i’). This proves that satisfies , for each . Since and , we obtain , for each , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is odd, since . Thus, Lemma 3 proves that is admissible. The final result is then a direct consequence of Theorem 6.", "index": 5}, {"type": "text", "bbox": [109, 587, 500, 617], "content": "We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].", "index": 6}, {"type": "text", "bbox": [109, 618, 500, 674], "content": "An interesting problem which naturally arises is that of characterizing the set of branching knots in involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers and , with odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot of type , then can be assumed to be even.", "index": 7}]
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[{"bbox": [109, 128, 500, 142], "content": "Since the handlebody orbifolds and their gluing only depend on , the", "parent_index": 0, "line_index": 0}, {"bbox": [109, 142, 500, 156], "content": "same holds for the branching set . The homeomorphism type of follows", "parent_index": 0, "line_index": 1}, {"bbox": [110, 158, 302, 169], "content": "from Proposition 2 and Lemma 5.", "parent_index": 0, "line_index": 2}, {"bbox": [109, 176, 500, 191], "content": "Remark 4. More generally, given two positive integers and such", "parent_index": 1, "line_index": 0}, {"bbox": [109, 190, 500, 205], "content": "that divides , if is admissible, then the Dunwoody man-", "parent_index": 1, "line_index": 1}, {"bbox": [109, 205, 500, 219], "content": "ifold is the -fold cyclic covering of the manifold", "parent_index": 1, "line_index": 2}, {"bbox": [110, 219, 388, 234], "content": ", branched over an -knot in .", "parent_index": 1, "line_index": 3}, {"bbox": [110, 239, 499, 254], "content": "Example 2. The Dunwoody manifolds ,", "parent_index": 2, "line_index": 0}, {"bbox": [110, 254, 500, 268], "content": "and , with coprime, are -fold cyclic coverings of the", "parent_index": 2, "line_index": 1}, {"bbox": [109, 268, 500, 283], "content": "manifolds , and respectively, branched over a trivial knot.", "parent_index": 2, "line_index": 2}, {"bbox": [109, 282, 499, 297], "content": "In fact, these Dunwoody manifolds are the connected sum of copies of ,", "parent_index": 2, "line_index": 3}, {"bbox": [110, 296, 276, 312], "content": "and respectively.", "parent_index": 2, "line_index": 4}, {"bbox": [126, 316, 499, 331], "content": "Let us consider now the class of the Dunwoody manifolds", "parent_index": 3, "line_index": 0}, {"bbox": [110, 331, 500, 347], "content": "with (and hence odd) and . Many ex-", "parent_index": 3, "line_index": 1}, {"bbox": [111, 347, 499, 360], "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "parent_index": 3, "line_index": 2}, {"bbox": [109, 361, 501, 375], "content": "that they are -fold cyclic coverings of , branched over suitable knots. The", "parent_index": 3, "line_index": 3}, {"bbox": [109, 375, 395, 389], "content": "following corollary of Theorem 6 proves this conjecture.", "parent_index": 3, "line_index": 4}, {"bbox": [109, 399, 499, 416], "content": "Corollary 7 Let be an admissible 6-tuple with", "parent_index": 4, "line_index": 0}, {"bbox": [109, 414, 502, 431], "content": "and . Then the -tuple is admissible for each", "parent_index": 4, "line_index": 1}, {"bbox": [110, 429, 501, 444], "content": "and the Dunwoody manifold is a n-fold cyclic", "parent_index": 4, "line_index": 2}, {"bbox": [110, 443, 501, 456], "content": "coverings of , branched over a genus one -bridge knot , which is", "parent_index": 4, "line_index": 3}, {"bbox": [110, 458, 204, 473], "content": "independent on .", "parent_index": 4, "line_index": 4}, {"bbox": [126, 483, 500, 497], "content": "Proof. Obviously . Since is admissible, it satisfies", "parent_index": 5, "line_index": 0}, {"bbox": [110, 496, 500, 514], "content": "(i’). This proves that satisfies , for each . Since", "parent_index": 5, "line_index": 1}, {"bbox": [110, 513, 500, 527], "content": "and , we obtain , for each , which", "parent_index": 5, "line_index": 2}, {"bbox": [110, 526, 500, 541], "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is", "parent_index": 5, "line_index": 3}, {"bbox": [109, 541, 501, 556], "content": "odd, since . Thus, Lemma", "parent_index": 5, "line_index": 4}, {"bbox": [110, 556, 499, 569], "content": "3 proves that is admissible. The final result is then a direct consequence", "parent_index": 5, "line_index": 5}, {"bbox": [110, 569, 200, 583], "content": "of Theorem 6.", "parent_index": 5, "line_index": 6}, {"bbox": [127, 589, 499, 604], "content": "We point out that the above result has been independently obtained by", "parent_index": 6, "line_index": 0}, {"bbox": [110, 605, 281, 618], "content": "H. J. Song and S. H. Kim in [32].", "parent_index": 6, "line_index": 1}, {"bbox": [127, 618, 500, 633], "content": "An interesting problem which naturally arises is that of characterizing the", "parent_index": 7, "line_index": 0}, {"bbox": [109, 633, 500, 648], "content": "set of branching knots in involved in Corollary 7. The next theorem", "parent_index": 7, "line_index": 1}, {"bbox": [110, 649, 500, 661], "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "parent_index": 7, "line_index": 2}, {"bbox": [109, 661, 501, 676], "content": "is determined by two coprime integers and , with odd. The", "parent_index": 7, "line_index": 3}, {"bbox": [110, 128, 500, 141], "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "parent_index": 7, "line_index": 4}, {"bbox": [111, 142, 500, 156], "content": "[30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot", "parent_index": 7, "line_index": 5}, {"bbox": [110, 157, 387, 170], "content": "of type , then can be assumed to be even.", "parent_index": 7, "line_index": 6}]
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[]
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[{"bbox": [438, 129, 476, 141], "content": "a,b,c,r", "parent_index": 0, "subtype": "inline"}, {"bbox": [277, 144, 288, 153], "content": "K", "parent_index": 0, "subtype": "inline"}, {"bbox": [446, 144, 461, 153], "content": "M^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [422, 181, 429, 187], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [460, 178, 470, 187], "content": "n^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [136, 192, 146, 201], "content": "n^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [191, 196, 198, 201], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [218, 191, 286, 204], "content": "(a,b,c,r,n,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [137, 206, 218, 218], "content": "M(a,b,c,n,r,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [256, 206, 279, 218], "content": "n/n^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [470, 206, 500, 218], "content": "M^{\\prime}=", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 221, 195, 233], "content": "M(a,b,c,n^{\\prime},r,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [293, 221, 324, 234], "content": "(n^{\\prime},1)", "parent_index": 1, "subtype": "inline"}, {"bbox": [368, 221, 383, 230], "content": "M^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [322, 241, 407, 253], "content": "M(0,0,1,n,0,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [414, 241, 499, 253], "content": "M(1,0,0,n,1,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [135, 255, 217, 267], "content": "M(0,0,c,n,r,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [254, 259, 270, 267], "content": "c,r", "parent_index": 2, "subtype": "inline"}, {"bbox": [346, 259, 353, 264], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [163, 269, 176, 279], "content": "\\mathbf{S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [183, 269, 223, 280], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [250, 270, 282, 282], "content": "L(c,r)", "parent_index": 2, "subtype": "inline"}, {"bbox": [424, 288, 432, 294], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [483, 284, 496, 294], "content": "\\mathbf{S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 298, 150, 308], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [177, 298, 210, 311], "content": "L(c,r)", "parent_index": 2, "subtype": "inline"}, {"bbox": [463, 318, 499, 330], "content": "\\textstyle M_{n}\\ =", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 333, 192, 345], "content": "M(a,b,c,n,r,s)", "parent_index": 3, "subtype": "inline"}, {"bbox": [224, 334, 264, 345], "content": "p=\\pm1", "parent_index": 3, "subtype": "inline"}, {"bbox": [330, 334, 337, 342], "content": "d", "parent_index": 3, "subtype": "inline"}, {"bbox": [393, 335, 439, 345], "content": "s\\,=\\,-p q", "parent_index": 3, "subtype": "inline"}, {"bbox": [179, 366, 186, 371], "content": "n", "parent_index": 3, "subtype": "inline"}, {"bbox": [306, 361, 319, 371], "content": "\\mathbf{S^{3}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [203, 401, 298, 414], "content": "\\sigma_{1}=(a,b,c,1,r,0)", "parent_index": 4, "subtype": "inline"}, {"bbox": [452, 402, 499, 414], "content": "p_{\\sigma_{1}}=\\pm1", "parent_index": 4, "subtype": "inline"}, {"bbox": [132, 417, 194, 428], "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [252, 417, 258, 425], "content": "\\it6", "parent_index": 4, "subtype": "inline"}, {"bbox": [289, 415, 387, 428], "content": "\\sigma_{n}=(a,b,c,n,r,s)", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 431, 140, 440], "content": "n>1", "parent_index": 4, "subtype": "inline"}, {"bbox": [291, 430, 408, 443], "content": "M_{n}=M(a,b,c,n,r,s)", "parent_index": 4, "subtype": "inline"}, {"bbox": [175, 444, 189, 454], "content": "\\mathrm{{S^{3}}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [338, 445, 343, 454], "content": "\\mathit{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [408, 444, 449, 455], "content": "K\\subset{\\bf S^{3}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [191, 463, 199, 469], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [225, 484, 321, 497], "content": "(a,b,c,1,r,s)=\\sigma_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [360, 488, 371, 496], "content": "\\sigma_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [225, 503, 237, 510], "content": "\\sigma_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [285, 498, 300, 511], "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "parent_index": 5, "subtype": "inline"}, {"bbox": [353, 500, 383, 509], "content": "n>1", "parent_index": 5, "subtype": "inline"}, {"bbox": [423, 500, 500, 511], "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 515, 151, 525], "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [176, 514, 255, 525], "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "parent_index": 5, "subtype": "inline"}, {"bbox": [314, 514, 385, 525], "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [434, 514, 463, 523], "content": "n>1", "parent_index": 5, "subtype": "inline"}, {"bbox": [481, 528, 488, 537], "content": "d", "parent_index": 5, "subtype": "inline"}, {"bbox": [165, 542, 420, 555], "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [182, 560, 194, 568], "content": "\\sigma_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [129, 635, 138, 644], "content": "\\kappa", "parent_index": 7, "subtype": "inline"}, {"bbox": [258, 634, 271, 644], "content": "\\mathrm{{S^{3}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [320, 667, 328, 672], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [358, 664, 366, 675], "content": "\\beta", "parent_index": 7, "subtype": "inline"}, {"bbox": [404, 664, 440, 673], "content": "\\alpha~>~0", "parent_index": 7, "subtype": "inline"}, {"bbox": [298, 143, 326, 155], "content": "(\\alpha,\\beta)", "parent_index": 7, "subtype": "inline"}, {"bbox": [150, 158, 201, 170], "content": "(\\alpha,\\alpha-\\beta)", "parent_index": 7, "subtype": "inline"}, {"bbox": [235, 158, 243, 169], "content": "\\beta", "parent_index": 7, "subtype": "inline"}]
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Theorem 8 The $\it6$ -tuple $\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\!+\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\sigma_{1}}$ , then the $\it6$ -tuple $\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\!+\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.
Proof. From $(2a+1,2r)=1$ it immediately follows that $\sigma_{1}$ has a unique cycle in $\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\sigma_{1}$ is admissible. Since $p_{\sigma_{n}}~=~p_{\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over a knot $K\subset{\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\;=\;2$ . Note that $[s]_{2}\,=$ $[-q_{\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\,=\,2$ we can suppose $s\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\mathrm{{S^{3}}}$ is unique [14], the result immediately holds.
Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24].
An immediate consequence of Theorem 8 is:
Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.
Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.
About the set $\kappa$ of knots in $\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following:
Conjecture. The set $\kappa$ contains all torus knots.
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<p data-bbox="109 180 500 253">Theorem 8 The $\it6$ -tuple $\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\!+\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\sigma_{1}}$ , then the $\it6$ -tuple $\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\!+\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds. </p>
<p data-bbox="109 266 500 453">Proof. From $(2a+1,2r)=1$ it immediately follows that $\sigma_{1}$ has a unique cycle in $\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\sigma_{1}$ is admissible. Since $p_{\sigma_{n}}~=~p_{\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over a knot $K\subset{\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\;=\;2$ . Note that $[s]_{2}\,=$ $[-q_{\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\,=\,2$ we can suppose $s\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\mathrm{{S^{3}}}$ is unique [14], the result immediately holds. </p>
<p data-bbox="110 459 500 503">Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24]. </p>
<p data-bbox="127 508 354 522">An immediate consequence of Theorem 8 is: </p>
<p data-bbox="110 535 500 563">Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric. </p>
<p data-bbox="110 576 500 619">Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric. </p>
<p data-bbox="109 625 501 654">About the set $\kappa$ of knots in $\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following: </p>
<p data-bbox="110 660 362 674">Conjecture. The set $\kappa$ contains all torus knots. </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 12}, {"type": "text", "text": "Theorem 8 The $\\it6$ -tuple $\\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\\!+\\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\\sigma_{1}}$ , then the $\\it6$ -tuple $\\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\\!+\\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds. ", "page_idx": 12}, {"type": "text", "text": "Proof. From $(2a+1,2r)=1$ it immediately follows that $\\sigma_{1}$ has a unique cycle in $\\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\\sigma_{1}$ is admissible. Since $p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over a knot $K\\subset{\\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\\;=\\;2$ . Note that $[s]_{2}\\,=$ $[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\\,=\\,2$ we can suppose $s\\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\\mathrm{{S^{3}}}$ is unique [14], the result immediately holds. ", "page_idx": 12}, {"type": "text", "text": "Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24]. ", "page_idx": 12}, {"type": "text", "text": "An immediate consequence of Theorem 8 is: ", "page_idx": 12}, {"type": "text", "text": "Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric. ", "page_idx": 12}, {"type": "text", "text": "Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric. ", "page_idx": 12}, {"type": "text", "text": "About the set $\\kappa$ of knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following: ", "page_idx": 12}, {"type": "text", "text": "Conjecture. The set $\\kappa$ contains all torus knots. ", "page_idx": 12}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [110, 128, 500, 141], "spans": [{"bbox": [110, 128, 500, 141], "score": 1.0, "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "type": "text"}], "index": 0}, {"bbox": [111, 142, 500, 156], "spans": [{"bbox": [111, 142, 297, 156], "score": 1.0, "content": "[30]. Since the 2-bridge knot of type ", "type": "text"}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text"}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text"}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3–10 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "bbox_fs": [110, 128, 500, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5, "bbox_fs": [109, 181, 501, 256]}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3–10 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14, "bbox_fs": [109, 267, 501, 455]}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [109, 461, 500, 505]}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [128, 510, 354, 524]}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [110, 537, 501, 565]}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28, "bbox_fs": [109, 577, 500, 621]}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [109, 626, 500, 657]}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32, "bbox_fs": [111, 662, 362, 675]}]}
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[{"type": "text", "bbox": [110, 125, 500, 168], "content": "", "index": 0}, {"type": "text", "bbox": [109, 180, 500, 253], "content": "Theorem 8 The -tuple with is admissi- ble. Moreover, if , then the -tuple is admissible for each and the Dunwoody manifold is the - fold cyclic covering of , branched over the 2-bridge knot of type . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "index": 1}, {"type": "text", "bbox": [109, 266, 500, 453], "content": "Proof. From it immediately follows that has a unique cycle in . Since is odd, Corollary 4 proves that is admissible. Since , all assumptions of Corollary 7 hold; hence is admissible for each and is an -fold cyclic covering of , branched over a knot which is independent on . In order to determine this knot, we can restrict our attention to the case . Note that and hence is always even. Thus, in the case we can suppose . Let us consider now the genus two Heegaard diagram . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space (see Figure 10). Since the representation of lens spaces (including ) as 2-fold branched coverings of is unique [14], the result immediately holds.", "index": 2}, {"type": "text", "bbox": [110, 459, 500, 503], "content": "Remark 5. The Dunwoody manifold of Theorem 8 is home- omorphic to the Minkus manifold [21] and the Lins-Mandel manifold [19, 24].", "index": 3}, {"type": "text", "bbox": [127, 508, 354, 522], "content": "An immediate consequence of Theorem 8 is:", "index": 4}, {"type": "text", "bbox": [110, 535, 500, 563], "content": "Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.", "index": 5}, {"type": "text", "bbox": [110, 576, 500, 619], "content": "Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.", "index": 6}, {"type": "text", "bbox": [109, 625, 501, 654], "content": "About the set of knots in involved in Corollary 7, we propose the following:", "index": 7}, {"type": "text", "bbox": [110, 660, 362, 674], "content": "Conjecture. The set contains all torus knots.", "index": 8}]
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[{"bbox": [109, 181, 501, 198], "content": "Theorem 8 The -tuple with is admissi-", "parent_index": 1, "line_index": 0}, {"bbox": [109, 196, 501, 214], "content": "ble. Moreover, if , then the -tuple is admissible", "parent_index": 1, "line_index": 1}, {"bbox": [109, 212, 501, 227], "content": "for each and the Dunwoody manifold is the -", "parent_index": 1, "line_index": 2}, {"bbox": [109, 226, 500, 241], "content": "fold cyclic covering of , branched over the 2-bridge knot of type .", "parent_index": 1, "line_index": 3}, {"bbox": [110, 240, 499, 256], "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "parent_index": 1, "line_index": 4}, {"bbox": [126, 267, 500, 281], "content": "Proof. From it immediately follows that has a unique", "parent_index": 2, "line_index": 0}, {"bbox": [110, 282, 499, 296], "content": "cycle in . Since is odd, Corollary 4 proves that is admissible.", "parent_index": 2, "line_index": 1}, {"bbox": [109, 296, 501, 312], "content": "Since , all assumptions of Corollary 7 hold; hence is", "parent_index": 2, "line_index": 2}, {"bbox": [110, 311, 501, 326], "content": "admissible for each and is an -fold cyclic covering of , branched", "parent_index": 2, "line_index": 3}, {"bbox": [109, 325, 500, 339], "content": "over a knot which is independent on . In order to determine this", "parent_index": 2, "line_index": 4}, {"bbox": [109, 340, 500, 353], "content": "knot, we can restrict our attention to the case . Note that", "parent_index": 2, "line_index": 5}, {"bbox": [110, 354, 501, 369], "content": "and hence is always even. Thus, in the case we", "parent_index": 2, "line_index": 6}, {"bbox": [109, 369, 500, 384], "content": "can suppose . Let us consider now the genus two Heegaard diagram", "parent_index": 2, "line_index": 7}, {"bbox": [110, 383, 500, 398], "content": ". The sequence of Singer moves [31] on this diagram, drawn", "parent_index": 2, "line_index": 8}, {"bbox": [110, 398, 500, 412], "content": "in Figures 3–10 and described in the Appendix of the paper, leads to the", "parent_index": 2, "line_index": 9}, {"bbox": [109, 412, 500, 426], "content": "canonical genus one Heegaard diagram of the lens space (see", "parent_index": 2, "line_index": 10}, {"bbox": [110, 426, 500, 441], "content": "Figure 10). Since the representation of lens spaces (including ) as 2-fold", "parent_index": 2, "line_index": 11}, {"bbox": [109, 440, 482, 455], "content": "branched coverings of is unique [14], the result immediately holds.", "parent_index": 2, "line_index": 12}, {"bbox": [109, 461, 500, 476], "content": "Remark 5. The Dunwoody manifold of Theorem 8 is home-", "parent_index": 3, "line_index": 0}, {"bbox": [110, 476, 500, 489], "content": "omorphic to the Minkus manifold [21] and the Lins-Mandel", "parent_index": 3, "line_index": 1}, {"bbox": [110, 490, 293, 505], "content": "manifold [19, 24].", "parent_index": 3, "line_index": 2}, {"bbox": [128, 510, 354, 524], "content": "An immediate consequence of Theorem 8 is:", "parent_index": 4, "line_index": 0}, {"bbox": [110, 537, 501, 551], "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "parent_index": 5, "line_index": 0}, {"bbox": [111, 551, 428, 565], "content": "2-bridge knot admits a cyclic presentation which is geometric.", "parent_index": 5, "line_index": 1}, {"bbox": [109, 577, 500, 592], "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "parent_index": 6, "line_index": 0}, {"bbox": [110, 594, 500, 606], "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "parent_index": 6, "line_index": 1}, {"bbox": [110, 608, 357, 621], "content": "pointing out that this presentation is geometric.", "parent_index": 6, "line_index": 2}, {"bbox": [127, 626, 500, 641], "content": "About the set of knots in involved in Corollary 7, we propose the", "parent_index": 7, "line_index": 0}, {"bbox": [109, 641, 160, 657], "content": "following:", "parent_index": 7, "line_index": 1}, {"bbox": [111, 662, 362, 675], "content": "Conjecture. The set contains all torus knots.", "parent_index": 8, "line_index": 0}]
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[]
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[{"bbox": [204, 185, 210, 194], "content": "\\it6", "parent_index": 1, "subtype": "inline"}, {"bbox": [240, 184, 338, 196], "content": "\\sigma_{1}=(a,0,1,1,r,0)", "parent_index": 1, "subtype": "inline"}, {"bbox": [366, 184, 442, 197], "content": "(2a\\!+\\!1,2r)=1", "parent_index": 1, "subtype": "inline"}, {"bbox": [198, 200, 243, 211], "content": "s=-q_{\\sigma_{1}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [295, 200, 301, 208], "content": "\\it6", "parent_index": 1, "subtype": "inline"}, {"bbox": [333, 198, 432, 211], "content": "\\sigma_{n}=(a,0,1,n,r,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [155, 214, 184, 223], "content": "n>1", "parent_index": 1, "subtype": "inline"}, {"bbox": [335, 213, 451, 225], "content": "M_{n}=M(a,0,1,n,r,s)", "parent_index": 1, "subtype": "inline"}, {"bbox": [487, 217, 495, 223], "content": "n", "parent_index": 1, "subtype": "inline"}, {"bbox": [223, 227, 237, 237], "content": "\\mathrm{{S^{3}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [441, 226, 496, 240], "content": "(2a\\!+\\!1,2r)", "parent_index": 1, "subtype": "inline"}, {"bbox": [199, 269, 277, 281], "content": "(2a+1,2r)=1", "parent_index": 2, "subtype": "inline"}, {"bbox": [420, 272, 432, 280], "content": "\\sigma_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 284, 162, 293], "content": "\\mathcal{D}", "parent_index": 2, "subtype": "inline"}, {"bbox": [200, 284, 253, 294], "content": "d=2a+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [416, 287, 428, 294], "content": "\\sigma_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [142, 299, 232, 310], "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [473, 302, 485, 309], "content": "\\sigma_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [209, 313, 238, 322], "content": "n>1", "parent_index": 2, "subtype": "inline"}, {"bbox": [263, 313, 280, 323], "content": "M_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [310, 316, 317, 321], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [433, 312, 446, 322], "content": "\\mathrm{{S^{3}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [173, 326, 214, 336], "content": "K\\subset{\\bf S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [348, 330, 355, 336], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [363, 342, 398, 350], "content": "n\\;=\\;2", "parent_index": 2, "subtype": "inline"}, {"bbox": [467, 341, 500, 353], "content": "[s]_{2}\\,=", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 356, 203, 368], "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [263, 359, 269, 365], "content": "s", "parent_index": 2, "subtype": "inline"}, {"bbox": [450, 357, 481, 365], "content": "n\\,=\\,2", "parent_index": 2, "subtype": "inline"}, {"bbox": [177, 371, 208, 380], "content": "s\\implies0", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 384, 191, 397], "content": "H(a,0,1,2,r,0)", "parent_index": 2, "subtype": "inline"}, {"bbox": [407, 413, 475, 426], "content": "L(2a+1,2r)", "parent_index": 2, "subtype": "inline"}, {"bbox": [433, 428, 446, 437], "content": "\\mathbf{S^{3}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [224, 442, 237, 451], "content": "\\mathrm{{S^{3}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [302, 462, 386, 475], "content": "M(a,0,1,n,r,s)", "parent_index": 3, "subtype": "inline"}, {"bbox": [289, 477, 365, 489], "content": "M_{n}(2a+1,2r)", "parent_index": 3, "subtype": "inline"}, {"bbox": [158, 491, 248, 504], "content": "S(n,2a+1,2r,1)", "parent_index": 3, "subtype": "inline"}, {"bbox": [204, 629, 213, 638], "content": "\\kappa", "parent_index": 7, "subtype": "inline"}, {"bbox": [277, 628, 290, 638], "content": "\\mathrm{{S^{3}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [226, 664, 235, 672], "content": "\\kappa", "parent_index": 8, "subtype": "inline"}]
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If this conjecture is true, the set $\kappa$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ).
# 4 Appendix
Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III.
Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\,=\,0$ , the cycle $C_{1}^{\prime}$ (resp. $C_{2}^{\prime}$ ) is glued with the cycle $C_{1}^{\prime\prime}$ (resp. $C_{2}^{\prime\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) coming out from the vertex $v^{\prime}$ of $C_{1}^{\prime}$ (resp. $v^{\prime\prime}$ of $C_{2}^{\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\prime}$ with $C_{2}^{\prime}$ are oriented from $C_{1}^{\prime}$ to $C_{2}^{\prime}$ and all the $2a$ arcs connecting $C_{1}^{\prime\prime}$ with $C_{2}^{\prime\prime}$ are oriented from $C_{2}^{\prime\prime}$ to $C_{1}^{\prime\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\prime}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(2+2k)r$ .
The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\prime}=D_{1}\!+\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\prime}$ and $e^{\prime\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\prime}$ so that it becomes disjoint from $D_{2}^{\prime}$ .
The resulting Heegaard diagram is drawn in Figure 4. The new $2a+1$ pairs of vertices obtained on $C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime},C_{2}^{\prime\prime}$ are labelled by simply adding a prime to the old label, while the $4a+2$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $j^{\prime}$ is placed, in the cycles $C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ , between the old vertices labelled $j$ and $j+1$ respectively. The cycles $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ are no longer connected by any arc, while the cycles $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ are connected by a unique arc (belonging to $D_{1}^{\prime}$ ) joining the vertex labelled $(a+1)^{\prime}$ of $C_{1}^{\prime}$ with the vertex labelled $(a+1-r)^{\prime}$ of $C_{1}^{\prime\prime}$ . All the $\mathrm{3}a$ arcs connecting $C_{1}^{\prime}$ and $C_{2}^{\prime}$ are oriented from $C_{1}^{\prime}$ to $C_{2}^{\prime}$ and all the $\mathrm{3}a$ arcs which now connect $C_{1}^{\prime\prime}$ with $C_{2}^{\prime\prime}$ are oriented from $C_{2}^{\prime\prime}$ to $C_{1}^{\prime\prime}$ . The cycle $D_{2}^{\prime}$ contains exactly $4a+2$ arcs; more precisely, for each $i=1,\ldots,2a+1$ , it has one arc joining the vertex labelled $i$ of $C_{1}^{\prime}$ with the vertex labelled $2a+2-i$ of $C_{2}^{\prime}$ and one arc joining the vertex labelled $i$ of $C_{2}^{\prime\prime}$ with the vertex labelled $2a+2-2r-i$ of $C_{2}^{\prime}$ . The cycle $D_{1}^{\prime}$ is a copy of the cycle $D_{1}$ and hence it contains $2a+1$ arcs. One of these arcs connects $C_{1}^{\prime}$ with $C_{1}^{\prime\prime}$ ; moreover, for each $k=0,\ldots,a-1$ , $D_{1}^{\prime}$ has one arc joining the vertex of $C_{1}^{\prime}$ labelled $(a+1-(1+2k)r)^{\prime}$ with the vertex of $C_{2}^{\prime}$ labelled $(a+1+(1+2k)r)^{\prime}$ and one arc joining the vertex of $C_{2}^{\prime\prime}$ labelled $(a+1+(1+2k)r)^{\prime}$ with the vertex of $C_{1}^{\prime\prime}$ labelled $(a+1-(3+2k)r)^{\prime}$ .
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<p data-bbox="110 125 500 226">If this conjecture is true, the set $\kappa$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ). </p>
<h1 data-bbox="110 247 223 268">4 Appendix </h1>
<p data-bbox="110 278 500 350">Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III. </p>
<p data-bbox="110 351 500 581">Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\,=\,0$ , the cycle $C_{1}^{\prime}$ (resp. $C_{2}^{\prime}$ ) is glued with the cycle $C_{1}^{\prime\prime}$ (resp. $C_{2}^{\prime\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) coming out from the vertex $v^{\prime}$ of $C_{1}^{\prime}$ (resp. $v^{\prime\prime}$ of $C_{2}^{\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\prime}$ with $C_{2}^{\prime}$ are oriented from $C_{1}^{\prime}$ to $C_{2}^{\prime}$ and all the $2a$ arcs connecting $C_{1}^{\prime\prime}$ with $C_{2}^{\prime\prime}$ are oriented from $C_{2}^{\prime\prime}$ to $C_{1}^{\prime\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\prime}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(2+2k)r$ . </p>
<p data-bbox="110 582 500 654">The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\prime}=D_{1}\!+\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\prime}$ and $e^{\prime\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\prime}$ so that it becomes disjoint from $D_{2}^{\prime}$ . </p>
<p data-bbox="127 654 500 669">The resulting Heegaard diagram is drawn in Figure 4. The new $2a+1$ pairs of vertices obtained on $C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime},C_{2}^{\prime\prime}$ are labelled by simply adding a prime to the old label, while the $4a+2$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $j^{\prime}$ is placed, in the cycles $C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ , between the old vertices labelled $j$ and $j+1$ respectively. The cycles $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ are no longer connected by any arc, while the cycles $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ are connected by a unique arc (belonging to $D_{1}^{\prime}$ ) joining the vertex labelled $(a+1)^{\prime}$ of $C_{1}^{\prime}$ with the vertex labelled $(a+1-r)^{\prime}$ of $C_{1}^{\prime\prime}$ . All the $\mathrm{3}a$ arcs connecting $C_{1}^{\prime}$ and $C_{2}^{\prime}$ are oriented from $C_{1}^{\prime}$ to $C_{2}^{\prime}$ and all the $\mathrm{3}a$ arcs which now connect $C_{1}^{\prime\prime}$ with $C_{2}^{\prime\prime}$ are oriented from $C_{2}^{\prime\prime}$ to $C_{1}^{\prime\prime}$ . The cycle $D_{2}^{\prime}$ contains exactly $4a+2$ arcs; more precisely, for each $i=1,\ldots,2a+1$ , it has one arc joining the vertex labelled $i$ of $C_{1}^{\prime}$ with the vertex labelled $2a+2-i$ of $C_{2}^{\prime}$ and one arc joining the vertex labelled $i$ of $C_{2}^{\prime\prime}$ with the vertex labelled $2a+2-2r-i$ of $C_{2}^{\prime}$ . The cycle $D_{1}^{\prime}$ is a copy of the cycle $D_{1}$ and hence it contains $2a+1$ arcs. One of these arcs connects $C_{1}^{\prime}$ with $C_{1}^{\prime\prime}$ ; moreover, for each $k=0,\ldots,a-1$ , $D_{1}^{\prime}$ has one arc joining the vertex of $C_{1}^{\prime}$ labelled $(a+1-(1+2k)r)^{\prime}$ with the vertex of $C_{2}^{\prime}$ labelled $(a+1+(1+2k)r)^{\prime}$ and one arc joining the vertex of $C_{2}^{\prime\prime}$ labelled $(a+1+(1+2k)r)^{\prime}$ with the vertex of $C_{1}^{\prime\prime}$ labelled $(a+1-(3+2k)r)^{\prime}$ . </p>
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0003042v1
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[{"type": "text", "text": "If this conjecture is true, the set $\\kappa$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ). ", "page_idx": 13}, {"type": "text", "text": "4 Appendix ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III. ", "page_idx": 13}, {"type": "text", "text": "Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\\,=\\,0$ , the cycle $C_{1}^{\\prime}$ (resp. $C_{2}^{\\prime}$ ) is glued with the cycle $C_{1}^{\\prime\\prime}$ (resp. $C_{2}^{\\prime\\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ (resp. $v^{\\prime\\prime}$ of $C_{2}^{\\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\\prime}$ with $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $2a$ arcs connecting $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\\prime}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(2+2k)r$ . ", "page_idx": 13}, {"type": "text", "text": "The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\\prime}=D_{1}\\!+\\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\\prime}$ and $e^{\\prime\\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\\prime}$ so that it becomes disjoint from $D_{2}^{\\prime}$ . ", "page_idx": 13}, {"type": "text", "text": "The resulting Heegaard diagram is drawn in Figure 4. The new $2a+1$ pairs of vertices obtained on $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}$ are labelled by simply adding a prime to the old label, while the $4a+2$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $j^{\\prime}$ is placed, in the cycles $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ , between the old vertices labelled $j$ and $j+1$ respectively. The cycles $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ are no longer connected by any arc, while the cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are connected by a unique arc (belonging to $D_{1}^{\\prime}$ ) joining the vertex labelled $(a+1)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . All the $\\mathrm{3}a$ arcs connecting $C_{1}^{\\prime}$ and $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $\\mathrm{3}a$ arcs which now connect $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{2}^{\\prime}$ contains exactly $4a+2$ arcs; more precisely, for each $i=1,\\ldots,2a+1$ , it has one arc joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $2a+2-i$ of $C_{2}^{\\prime}$ and one arc joining the vertex labelled $i$ of $C_{2}^{\\prime\\prime}$ with the vertex labelled $2a+2-2r-i$ of $C_{2}^{\\prime}$ . The cycle $D_{1}^{\\prime}$ is a copy of the cycle $D_{1}$ and hence it contains $2a+1$ arcs. One of these arcs connects $C_{1}^{\\prime}$ with $C_{1}^{\\prime\\prime}$ ; moreover, for each $k=0,\\ldots,a-1$ , $D_{1}^{\\prime}$ has one arc joining the vertex of $C_{1}^{\\prime}$ labelled $(a+1-(1+2k)r)^{\\prime}$ with the vertex of $C_{2}^{\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ and one arc joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $(a+1-(3+2k)r)^{\\prime}$ . ", "page_idx": 13}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 500, 226], "lines": [{"bbox": [126, 127, 500, 142], "spans": [{"bbox": [126, 127, 291, 142], "score": 1.0, "content": "If this conjecture is true, the set ", "type": "text"}, {"bbox": [291, 129, 301, 138], "score": 0.92, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [301, 127, 500, 142], "score": 1.0, "content": " contains knots with an arbitrarily high", "type": "text"}], "index": 0}, {"bbox": [110, 143, 499, 156], "spans": [{"bbox": [110, 143, 499, 156], "score": 1.0, "content": "number of bridges. Moreover, the conjecture implies that every branched", "type": "text"}], "index": 1}, {"bbox": [110, 156, 500, 171], "spans": [{"bbox": [110, 156, 500, 171], "score": 1.0, "content": "cyclic covering of a torus knot admits a geometric cyclic presentation. The", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 501, 186], "score": 1.0, "content": "above conjecture is supported by several cases contained in Table 1 of [6]", "type": "text"}], "index": 3}, {"bbox": [110, 185, 499, 200], "spans": [{"bbox": [110, 185, 380, 200], "score": 1.0, "content": "(see [32]). For example, the Dunwoody manifolds ", "type": "text"}, {"bbox": [380, 186, 465, 199], "score": 0.92, "content": "M(1,2,3,n,4,4)", "type": "inline_equation", "height": 13, "width": 85}, {"bbox": [466, 185, 499, 200], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 201, 197, 213], "score": 0.91, "content": "M(1,3,4,n,5,5))", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [198, 200, 246, 214], "score": 1.0, "content": " are the ", "type": "text"}, {"bbox": [247, 205, 254, 210], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [255, 200, 500, 214], "score": 1.0, "content": "-fold branched cyclic coverings of the 4-bridge", "type": "text"}], "index": 5}, {"bbox": [109, 214, 423, 228], "spans": [{"bbox": [109, 214, 167, 228], "score": 1.0, "content": "torus knot ", "type": "text"}, {"bbox": [167, 215, 204, 228], "score": 0.94, "content": "K(4,5)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [205, 214, 378, 228], "score": 1.0, "content": " (resp. of the 5-bridge torus knot ", "type": "text"}, {"bbox": [378, 215, 415, 228], "score": 0.94, "content": "K(5,6)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [415, 214, 423, 228], "score": 1.0, "content": ").", "type": "text"}], "index": 6}], "index": 3}, {"type": "title", "bbox": [110, 247, 223, 268], "lines": [{"bbox": [111, 250, 223, 268], "spans": [{"bbox": [111, 253, 122, 264], "score": 1.0, "content": "4", "type": "text"}, {"bbox": [137, 250, 223, 268], "score": 1.0, "content": "Appendix", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [110, 278, 500, 350], "lines": [{"bbox": [109, 279, 500, 295], "spans": [{"bbox": [109, 279, 500, 295], "score": 1.0, "content": "Now we show how to obtain, by means of Singer moves [31] on the genus", "type": "text"}], "index": 8}, {"bbox": [109, 294, 501, 310], "spans": [{"bbox": [109, 294, 229, 310], "score": 1.0, "content": "two Heegaard diagram ", "type": "text"}, {"bbox": [230, 295, 311, 308], "score": 0.94, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [311, 294, 501, 310], "score": 1.0, "content": " of Figure 3, the canonical genus one", "type": "text"}], "index": 9}, {"bbox": [110, 310, 500, 323], "spans": [{"bbox": [110, 310, 299, 323], "score": 1.0, "content": "Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [299, 311, 367, 323], "score": 0.93, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [367, 310, 500, 323], "score": 1.0, "content": " of Figure 10. The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". Note that,", "type": "text"}], "index": 13}, {"bbox": [109, 367, 500, 382], "spans": [{"bbox": [109, 367, 139, 382], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [139, 370, 168, 378], "score": 0.91, "content": "s\\,=\\,0", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [169, 367, 226, 382], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [226, 369, 239, 380], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [239, 367, 279, 382], "score": 1.0, "content": " (resp. 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", "type": "text"}, {"bbox": [187, 383, 202, 394], "score": 0.84, "content": "D_{2}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [203, 382, 500, 396], "score": 1.0, "content": ") be the cycle of the Heegaard diagram corresponding to", "type": "text"}], "index": 15}, {"bbox": [109, 396, 499, 410], "spans": [{"bbox": [109, 396, 151, 410], "score": 1.0, "content": "the arc ", "type": "text"}, {"bbox": [151, 397, 160, 406], "score": 0.88, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [160, 396, 200, 410], "score": 1.0, "content": " (resp. 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The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10, "bbox_fs": [109, 279, 501, 352]}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". 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The new ", "type": "text"}, {"bbox": [466, 658, 499, 668], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 33}], "index": 34}, {"bbox": [109, 127, 501, 144], "spans": [{"bbox": [109, 127, 261, 144], "score": 1.0, "content": "pairs of vertices obtained on ", "type": "text", "cross_page": true}, {"bbox": [261, 129, 331, 141], "score": 0.94, "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 70, "cross_page": true}, {"bbox": [332, 127, 501, 144], "score": 1.0, "content": " are labelled by simply adding a", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 500, 156], "spans": [{"bbox": [110, 142, 289, 156], "score": 1.0, "content": "prime to the old label, while the ", "type": "text", "cross_page": true}, {"bbox": [289, 144, 324, 154], "score": 0.92, "content": "4a+2", "type": "inline_equation", "height": 10, "width": 35, "cross_page": true}, {"bbox": [324, 142, 500, 156], "score": 1.0, "content": " pairs of fixed vertices keep their", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 499, 171], "spans": [{"bbox": [110, 157, 369, 171], "score": 1.0, "content": "old labelling. 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All the", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 229, 501, 244], "spans": [{"bbox": [110, 231, 122, 240], "score": 0.89, "content": "\\mathrm{3}a", "type": "inline_equation", "height": 9, "width": 12, "cross_page": true}, {"bbox": [123, 229, 208, 244], "score": 1.0, "content": " arcs connecting ", "type": "text", "cross_page": true}, {"bbox": [208, 230, 221, 242], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [222, 229, 247, 244], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [248, 230, 261, 242], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [261, 229, 356, 244], "score": 1.0, "content": " are oriented from ", "type": "text", "cross_page": true}, {"bbox": [356, 230, 369, 242], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [370, 229, 387, 244], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [387, 230, 400, 242], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [401, 229, 462, 244], "score": 1.0, "content": " and all the ", "type": "text", "cross_page": true}, {"bbox": [463, 231, 475, 240], "score": 0.88, "content": "\\mathrm{3}a", "type": "inline_equation", "height": 9, "width": 12, "cross_page": true}, {"bbox": [475, 229, 501, 244], "score": 1.0, "content": " arcs", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [108, 242, 498, 259], "spans": [{"bbox": [108, 242, 212, 259], "score": 1.0, "content": "which now connect ", "type": "text", "cross_page": true}, {"bbox": [213, 245, 227, 257], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [227, 242, 258, 259], "score": 1.0, "content": " with ", "type": "text", "cross_page": true}, {"bbox": [259, 245, 273, 257], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [273, 242, 371, 259], "score": 1.0, "content": " are oriented from ", "type": "text", "cross_page": true}, {"bbox": [372, 245, 386, 257], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [387, 242, 405, 259], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [406, 245, 420, 257], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [420, 242, 484, 259], "score": 1.0, "content": ". The cycle ", "type": "text", "cross_page": true}, {"bbox": [484, 245, 498, 257], "score": 0.92, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 195, 272], "score": 1.0, "content": "contains exactly ", "type": "text", "cross_page": true}, {"bbox": [195, 260, 227, 269], "score": 0.92, "content": "4a+2", "type": "inline_equation", "height": 9, "width": 32, "cross_page": true}, {"bbox": [227, 258, 379, 272], "score": 1.0, "content": " arcs; more precisely, for each ", "type": "text", "cross_page": true}, {"bbox": [379, 260, 464, 271], "score": 0.92, "content": "i=1,\\ldots,2a+1", "type": "inline_equation", "height": 11, "width": 85, "cross_page": true}, {"bbox": [464, 258, 500, 272], "score": 1.0, "content": ", it has", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [110, 273, 498, 285], "spans": [{"bbox": [110, 273, 287, 285], "score": 1.0, "content": "one arc joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [288, 274, 291, 282], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 3, "cross_page": true}, {"bbox": [292, 273, 308, 285], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [308, 273, 321, 285], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [321, 273, 449, 285], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [450, 274, 498, 284], "score": 0.92, "content": "2a+2-i", "type": "inline_equation", "height": 10, "width": 48, "cross_page": true}], "index": 10}, {"bbox": [109, 287, 500, 300], "spans": [{"bbox": [109, 287, 123, 300], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 288, 136, 300], "score": 0.92, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [137, 287, 339, 300], "score": 1.0, "content": " and one arc joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [339, 289, 343, 297], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4, "cross_page": true}, {"bbox": [344, 287, 360, 300], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [360, 288, 374, 300], "score": 0.93, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [375, 287, 500, 300], "score": 1.0, "content": " with the vertex labelled", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [110, 300, 500, 316], "spans": [{"bbox": [110, 303, 191, 313], "score": 0.9, "content": "2a+2-2r-i", "type": "inline_equation", "height": 10, "width": 81, "cross_page": true}, {"bbox": [192, 300, 210, 316], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [210, 302, 224, 314], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [224, 300, 290, 316], "score": 1.0, "content": ". The cycle ", "type": "text", "cross_page": true}, {"bbox": [290, 302, 304, 314], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [305, 300, 426, 316], "score": 1.0, "content": " is a copy of the cycle ", "type": "text", "cross_page": true}, {"bbox": [427, 303, 441, 313], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [442, 300, 500, 316], "score": 1.0, "content": " and hence", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [107, 312, 500, 332], "spans": [{"bbox": [107, 312, 169, 332], "score": 1.0, "content": "it contains ", "type": "text", "cross_page": true}, {"bbox": [169, 317, 203, 327], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 34, "cross_page": true}, {"bbox": [203, 312, 381, 332], "score": 1.0, "content": " arcs. One of these arcs connects ", "type": "text", "cross_page": true}, {"bbox": [381, 317, 394, 329], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [395, 312, 426, 332], "score": 1.0, "content": " with ", "type": "text", "cross_page": true}, {"bbox": [426, 317, 441, 329], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [441, 312, 500, 332], "score": 1.0, "content": "; moreover,", "type": "text", "cross_page": true}], "index": 13}, {"bbox": [108, 329, 501, 344], "spans": [{"bbox": [108, 329, 155, 344], "score": 1.0, "content": "for each ", "type": "text", "cross_page": true}, {"bbox": [155, 331, 241, 343], "score": 0.92, "content": "k=0,\\ldots,a-1", "type": "inline_equation", "height": 12, "width": 86, "cross_page": true}, {"bbox": [241, 329, 249, 344], "score": 1.0, "content": ", ", "type": "text", "cross_page": true}, {"bbox": [249, 331, 263, 343], "score": 0.94, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [264, 329, 441, 344], "score": 1.0, "content": " has one arc joining the vertex of ", "type": "text", "cross_page": true}, {"bbox": [442, 331, 455, 343], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [455, 329, 501, 344], "score": 1.0, "content": " labelled", "type": "text", "cross_page": true}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 212, 358], "score": 0.93, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 102, "cross_page": true}, {"bbox": [213, 345, 313, 358], "score": 1.0, "content": " with the vertex of ", "type": "text", "cross_page": true}, {"bbox": [313, 346, 326, 358], "score": 0.92, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [326, 345, 375, 358], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [375, 345, 476, 358], "score": 0.92, "content": "(a+1+(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 101, "cross_page": true}, {"bbox": [476, 345, 500, 358], "score": 1.0, "content": " and", "type": "text", "cross_page": true}], "index": 15}, {"bbox": [109, 359, 500, 372], "spans": [{"bbox": [109, 359, 257, 372], "score": 1.0, "content": "one arc joining the vertex of ", "type": "text", "cross_page": true}, {"bbox": [258, 360, 272, 372], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [272, 359, 320, 372], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [320, 360, 417, 372], "score": 0.92, "content": "(a+1+(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 12, "width": 97, "cross_page": true}, {"bbox": [418, 359, 500, 372], "score": 1.0, "content": " with the vertex", "type": "text", "cross_page": true}], "index": 16}, {"bbox": [110, 372, 289, 387], "spans": [{"bbox": [110, 372, 123, 387], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 375, 138, 387], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [138, 372, 185, 387], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [186, 374, 285, 387], "score": 0.91, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 99, "cross_page": true}, {"bbox": [286, 372, 289, 387], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 17}], "index": 34, "bbox_fs": [127, 656, 499, 671]}]}
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[{"type": "text", "bbox": [110, 125, 500, 226], "content": "If this conjecture is true, the set contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds (resp. are the -fold branched cyclic coverings of the 4-bridge torus knot (resp. of the 5-bridge torus knot ).", "index": 0}, {"type": "title", "bbox": [110, 247, 223, 268], "content": "4 Appendix", "index": 1}, {"type": "text", "bbox": [110, 278, 500, 350], "content": "Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram of Figure 3, the canonical genus one Heegaard diagram of the lens space of Figure 10. The result will be achieved by a sequence of exactly Singer moves: one of type ID, of type IC and the final one of type III.", "index": 2}, {"type": "text", "bbox": [110, 351, 500, 581], "content": "Figure 3 shows the open Heegaard diagram . Note that, since , the cycle (resp. ) is glued with the cycle (resp. ). Let (resp. ) be the cycle of the Heegaard diagram corresponding to the arc (resp. ) coming out from the vertex of (resp. of ) labelled . Orient (resp. ) so that the arc (resp. ) is oriented from up to down (resp. from down to up). This orientation on is opposite to the canonical one but, in this way, all the arcs connecting with are oriented from to and all the arcs connecting with are oriented from to . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , and the other one joining the vertex of labelled with the vertex of labelled . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , the other joining the vertex of labelled with the vertex of labelled .", "index": 3}, {"type": "text", "bbox": [110, 582, 500, 654], "content": "The first Singer move consists of replacing the curve with the curve (move of type ID of [31]) obtained by isotopically approaching the arcs and until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, in so that it becomes disjoint from .", "index": 4}, {"type": "text", "bbox": [127, 654, 500, 669], "content": "The resulting Heegaard diagram is drawn in Figure 4. The new pairs of vertices obtained on are labelled by simply adding a prime to the old label, while the pairs of fixed vertices keep their old labelling. Note that each new vertex labelled is placed, in the cycles and , between the old vertices labelled and respectively. The cycles and are no longer connected by any arc, while the cycles and are connected by a unique arc (belonging to ) joining the vertex labelled of with the vertex labelled of . All the arcs connecting and are oriented from to and all the arcs which now connect with are oriented from to . The cycle contains exactly arcs; more precisely, for each , it has one arc joining the vertex labelled of with the vertex labelled of and one arc joining the vertex labelled of with the vertex labelled of . The cycle is a copy of the cycle and hence it contains arcs. One of these arcs connects with ; moreover, for each , has one arc joining the vertex of labelled with the vertex of labelled and one arc joining the vertex of labelled with the vertex of labelled .", "index": 5}]
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[{"bbox": [126, 127, 500, 142], "content": "If this conjecture is true, the set contains knots with an arbitrarily high", "parent_index": 0, "line_index": 0}, {"bbox": [110, 143, 499, 156], "content": "number of bridges. Moreover, the conjecture implies that every branched", "parent_index": 0, "line_index": 1}, {"bbox": [110, 156, 500, 171], "content": "cyclic covering of a torus knot admits a geometric cyclic presentation. The", "parent_index": 0, "line_index": 2}, {"bbox": [109, 171, 501, 186], "content": "above conjecture is supported by several cases contained in Table 1 of [6]", "parent_index": 0, "line_index": 3}, {"bbox": [110, 185, 499, 200], "content": "(see [32]). For example, the Dunwoody manifolds (resp.", "parent_index": 0, "line_index": 4}, {"bbox": [110, 200, 500, 214], "content": "are the -fold branched cyclic coverings of the 4-bridge", "parent_index": 0, "line_index": 5}, {"bbox": [109, 214, 423, 228], "content": "torus knot (resp. of the 5-bridge torus knot ).", "parent_index": 0, "line_index": 6}, {"bbox": [111, 250, 223, 268], "content": "4 Appendix", "parent_index": 1, "line_index": 0}, {"bbox": [109, 279, 500, 295], "content": "Now we show how to obtain, by means of Singer moves [31] on the genus", "parent_index": 2, "line_index": 0}, {"bbox": [109, 294, 501, 310], "content": "two Heegaard diagram of Figure 3, the canonical genus one", "parent_index": 2, "line_index": 1}, {"bbox": [110, 310, 500, 323], "content": "Heegaard diagram of the lens space of Figure 10. The result", "parent_index": 2, "line_index": 2}, {"bbox": [110, 325, 499, 338], "content": "will be achieved by a sequence of exactly Singer moves: one of type ID,", "parent_index": 2, "line_index": 3}, {"bbox": [110, 338, 343, 352], "content": "of type IC and the final one of type III.", "parent_index": 2, "line_index": 4}, {"bbox": [128, 353, 499, 367], "content": "Figure 3 shows the open Heegaard diagram . Note that,", "parent_index": 3, "line_index": 0}, {"bbox": [109, 367, 500, 382], "content": "since , the cycle (resp. ) is glued with the cycle (resp. ).", "parent_index": 3, "line_index": 1}, {"bbox": [110, 382, 500, 396], "content": "Let (resp. ) be the cycle of the Heegaard diagram corresponding to", "parent_index": 3, "line_index": 2}, {"bbox": [109, 396, 499, 410], "content": "the arc (resp. ) coming out from the vertex of (resp. of )", "parent_index": 3, "line_index": 3}, {"bbox": [109, 410, 500, 424], "content": "labelled . Orient (resp. ) so that the arc (resp. ) is oriented", "parent_index": 3, "line_index": 4}, {"bbox": [109, 425, 500, 439], "content": "from up to down (resp. from down to up). This orientation on is opposite", "parent_index": 3, "line_index": 5}, {"bbox": [109, 438, 498, 454], "content": "to the canonical one but, in this way, all the arcs connecting with", "parent_index": 3, "line_index": 6}, {"bbox": [109, 454, 501, 468], "content": "are oriented from to and all the arcs connecting with are", "parent_index": 3, "line_index": 7}, {"bbox": [108, 468, 501, 484], "content": "oriented from to . The cycle , besides the arc , has two arcs for", "parent_index": 3, "line_index": 8}, {"bbox": [109, 483, 499, 498], "content": "each , one joining the vertex of labelled", "parent_index": 3, "line_index": 9}, {"bbox": [109, 497, 500, 512], "content": "with the vertex of labelled , and the other one joining", "parent_index": 3, "line_index": 10}, {"bbox": [110, 512, 500, 525], "content": "the vertex of labelled with the vertex of labelled", "parent_index": 3, "line_index": 11}, {"bbox": [110, 527, 500, 540], "content": ". The cycle , besides the arc , has two arcs for each", "parent_index": 3, "line_index": 12}, {"bbox": [110, 540, 500, 555], "content": ", one joining the vertex of labelled with", "parent_index": 3, "line_index": 13}, {"bbox": [109, 555, 499, 569], "content": "the vertex of labelled , the other joining the vertex of", "parent_index": 3, "line_index": 14}, {"bbox": [109, 569, 465, 583], "content": "labelled with the vertex of labelled .", "parent_index": 3, "line_index": 15}, {"bbox": [127, 584, 500, 597], "content": "The first Singer move consists of replacing the curve with the curve", "parent_index": 4, "line_index": 0}, {"bbox": [110, 598, 500, 613], "content": "(move of type ID of [31]) obtained by isotopically approaching", "parent_index": 4, "line_index": 1}, {"bbox": [109, 612, 500, 627], "content": "the arcs and until their intersection becomes a small arc and by removing", "parent_index": 4, "line_index": 2}, {"bbox": [109, 627, 501, 641], "content": "the interior of this arc. The move is completed by shifting, with a small", "parent_index": 4, "line_index": 3}, {"bbox": [109, 641, 389, 657], "content": "isotopy, in so that it becomes disjoint from .", "parent_index": 4, "line_index": 4}, {"bbox": [127, 656, 499, 671], "content": "The resulting Heegaard diagram is drawn in Figure 4. The new", "parent_index": 5, "line_index": 0}, {"bbox": [109, 127, 501, 144], "content": "pairs of vertices obtained on are labelled by simply adding a", "parent_index": 5, "line_index": 1}, {"bbox": [110, 142, 500, 156], "content": "prime to the old label, while the pairs of fixed vertices keep their", "parent_index": 5, "line_index": 2}, {"bbox": [110, 157, 499, 171], "content": "old labelling. Note that each new vertex labelled is placed, in the cycles", "parent_index": 5, "line_index": 3}, {"bbox": [110, 170, 500, 187], "content": "and , between the old vertices labelled and respectively.", "parent_index": 5, "line_index": 4}, {"bbox": [108, 184, 498, 201], "content": "The cycles and are no longer connected by any arc, while the cycles", "parent_index": 5, "line_index": 5}, {"bbox": [110, 200, 500, 214], "content": "and are connected by a unique arc (belonging to ) joining the vertex", "parent_index": 5, "line_index": 6}, {"bbox": [109, 213, 500, 228], "content": "labelled of with the vertex labelled of . All the", "parent_index": 5, "line_index": 7}, {"bbox": [110, 229, 501, 244], "content": "arcs connecting and are oriented from to and all the arcs", "parent_index": 5, "line_index": 8}, {"bbox": [108, 242, 498, 259], "content": "which now connect with are oriented from to . The cycle", "parent_index": 5, "line_index": 9}, {"bbox": [110, 258, 500, 272], "content": "contains exactly arcs; more precisely, for each , it has", "parent_index": 5, "line_index": 10}, {"bbox": [110, 273, 498, 285], "content": "one arc joining the vertex labelled of with the vertex labelled", "parent_index": 5, "line_index": 11}, {"bbox": [109, 287, 500, 300], "content": "of and one arc joining the vertex labelled of with the vertex labelled", "parent_index": 5, "line_index": 12}, {"bbox": [110, 300, 500, 316], "content": "of . The cycle is a copy of the cycle and hence", "parent_index": 5, "line_index": 13}, {"bbox": [107, 312, 500, 332], "content": "it contains arcs. One of these arcs connects with ; moreover,", "parent_index": 5, "line_index": 14}, {"bbox": [108, 329, 501, 344], "content": "for each , has one arc joining the vertex of labelled", "parent_index": 5, "line_index": 15}, {"bbox": [110, 345, 500, 358], "content": "with the vertex of labelled and", "parent_index": 5, "line_index": 16}, {"bbox": [109, 359, 500, 372], "content": "one arc joining the vertex of labelled with the vertex", "parent_index": 5, "line_index": 17}, {"bbox": [110, 372, 289, 387], "content": "of labelled .", "parent_index": 5, "line_index": 18}]
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[]
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"content": "v^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [397, 397, 410, 410], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [451, 397, 462, 406], "content": "v^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [481, 397, 494, 410], "content": "C_{2}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [154, 412, 180, 422], "content": "a+1", "parent_index": 3, "subtype": "inline"}, {"bbox": [225, 412, 240, 423], "content": "D_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [278, 412, 293, 423], "content": "D_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [380, 412, 389, 421], "content": "e^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [426, 412, 437, 421], "content": "e^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [427, 427, 442, 437], "content": "D_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [343, 442, 355, 450], "content": "2a", "parent_index": 3, "subtype": "inline"}, {"bbox": [442, 441, 455, 453], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [486, 441, 498, 453], "content": "C_{2}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [205, 455, 218, 467], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [237, 455, 251, 467], "content": "C_{2}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [317, 456, 329, 464], "content": "2a", "parent_index": 3, "subtype": "inline"}, {"bbox": [418, 455, 432, 467], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [464, 455, 478, 467], "content": "C_{2}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [184, 470, 199, 482], "content": "C_{2}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [218, 470, 232, 482], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [296, 470, 311, 481], "content": "D_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [400, 470, 408, 479], "content": "e^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [136, 484, 219, 495], "content": "k=0,\\dotsc,a-1", "parent_index": 3, "subtype": "inline"}, {"bbox": [353, 484, 366, 496], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [413, 483, 499, 496], "content": "a+1-(1+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [208, 498, 221, 510], "content": "C_{2}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [269, 498, 360, 511], "content": "a+1+(1+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [181, 513, 196, 525], "content": "C_{2}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [245, 513, 336, 525], "content": "a+1+(1+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [440, 513, 455, 525], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 527, 201, 540], "content": "a+1-(3+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [266, 528, 280, 538], "content": "D_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [370, 531, 381, 538], "content": "a_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 542, 192, 554], "content": "k=0,\\dotsc,a-1", "parent_index": 3, "subtype": "inline"}, {"bbox": [327, 542, 340, 554], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [387, 542, 473, 554], "content": "a+1-(2+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [178, 556, 191, 568], "content": "C_{2}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [237, 556, 321, 568], "content": "a+1+(2+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [484, 556, 499, 568], "content": "C_{2}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [153, 571, 212, 581], "content": "a+1+2k r", "parent_index": 3, "subtype": "inline"}, {"bbox": [312, 571, 326, 583], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [373, 570, 462, 583], "content": "a+1-(2+2k)r", "parent_index": 3, "subtype": "inline"}, {"bbox": [405, 586, 419, 596], "content": "D_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 600, 181, 612], "content": "D_{2}^{\\prime}=D_{1}\\!+\\!D_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [152, 614, 160, 623], "content": "e^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 614, 195, 623], "content": "e^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [153, 644, 167, 654], "content": "D_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 643, 199, 655], "content": "D_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [370, 643, 385, 655], "content": "D_{2}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [466, 658, 499, 668], "content": "2a+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [261, 129, 331, 141], "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [289, 144, 324, 154], "content": "4a+2", "parent_index": 5, "subtype": "inline"}, {"bbox": [369, 158, 378, 169], "content": "j^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 172, 161, 184], "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [187, 172, 201, 184], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [377, 173, 383, 184], "content": "j", "parent_index": 5, "subtype": "inline"}, {"bbox": [408, 173, 433, 184], "content": "j+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [166, 187, 179, 199], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [204, 187, 219, 199], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [486, 187, 498, 199], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [133, 201, 148, 213], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [384, 201, 399, 213], "content": "D_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [154, 215, 194, 228], "content": "(a+1)^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [212, 216, 225, 228], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [358, 215, 419, 228], "content": "(a+1-r)^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [438, 216, 452, 228], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 231, 122, 240], "content": "\\mathrm{3}a", "parent_index": 5, "subtype": "inline"}, {"bbox": [208, 230, 221, 242], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 230, 261, 242], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [356, 230, 369, 242], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [387, 230, 400, 242], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [463, 231, 475, 240], "content": "\\mathrm{3}a", "parent_index": 5, "subtype": "inline"}, {"bbox": [213, 245, 227, 257], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [259, 245, 273, 257], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [372, 245, 386, 257], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [406, 245, 420, 257], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [484, 245, 498, 257], "content": "D_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [195, 260, 227, 269], "content": "4a+2", "parent_index": 5, "subtype": "inline"}, {"bbox": [379, 260, 464, 271], "content": "i=1,\\ldots,2a+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [288, 274, 291, 282], "content": "i", "parent_index": 5, "subtype": "inline"}, {"bbox": [308, 273, 321, 285], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [450, 274, 498, 284], "content": "2a+2-i", "parent_index": 5, "subtype": "inline"}, {"bbox": [123, 288, 136, 300], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [339, 289, 343, 297], "content": "i", "parent_index": 5, "subtype": "inline"}, {"bbox": [360, 288, 374, 300], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 303, 191, 313], "content": "2a+2-2r-i", "parent_index": 5, "subtype": "inline"}, {"bbox": [210, 302, 224, 314], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [290, 302, 304, 314], "content": "D_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [427, 303, 441, 313], "content": "D_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [169, 317, 203, 327], "content": "2a+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [381, 317, 394, 329], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [426, 317, 441, 329], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [155, 331, 241, 343], "content": "k=0,\\ldots,a-1", "parent_index": 5, "subtype": "inline"}, {"bbox": [249, 331, 263, 343], "content": "D_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [442, 331, 455, 343], "content": "C_{1}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 345, 212, 358], "content": "(a+1-(1+2k)r)^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [313, 346, 326, 358], "content": "C_{2}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [375, 345, 476, 358], "content": "(a+1+(1+2k)r)^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [258, 360, 272, 372], "content": "C_{2}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [320, 360, 417, 372], "content": "(a+1+(1+2k)r)^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [123, 375, 138, 387], "content": "C_{1}^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [186, 374, 285, 387], "content": "(a+1-(3+2k)r)^{\\prime}", "parent_index": 5, "subtype": "inline"}]
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Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\prime\prime}$ and $C_{2}^{\prime\prime}$ and gluing the curve $C_{2}^{\prime\prime}$ of the resulting disc with $C_{2}^{\prime}$ .
The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\prime}$ and $E^{\prime\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ and they both have a unique vertex ( $w^{\prime}$ and $w^{\prime\prime}$ respectively). The cycle $E^{\prime}$ (resp. $E^{\prime\prime}$ ) is connected with $C_{1}^{\prime}$ (resp. with $C_{1}^{\prime\prime}$ ) by an arc joining $w^{\prime}$ (resp. $w^{\prime\prime}$ ) with the vertex labelled $(a+1)^{\prime}$ (resp. $(a+1-r)^{\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\prime}$ to $C_{1}^{\prime\prime}$ ; $2a+1$ of them belong to $D_{2}^{\prime}$ and the other $a$ belong to $D_{1}^{\prime}$ . More precisely, for each $i=1,\dots,2a+1$ , there is an arc of $D_{2}^{\prime}$ joining the vertex labelled $i$ of $C_{1}^{\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\prime\prime}$ ; while, for each $k=0,\dotsc,a-1$ , there is an arc of $D_{1}^{\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\prime}$ of $C_{1}^{\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\prime}$ of $C_{1}^{\prime\prime}$ .
Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $C_{1}^{\prime\prime}$ of the resulting disc with $C_{1}^{\prime}$ .
The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , which are copies of the cutting cycle $F_{1}$ . These cycles replace $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ and they both have one vertex less. It is easy to see that the cycle $D_{2}^{\prime}$ has exactly the same $2a+1$ arcs connecting $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , all oriented from $F_{1}^{\prime}$ to $F_{1}^{\prime\prime}$ ; if the labelling of the vertices of $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ is induced by the labelling of $F_{1}$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $D_{1}^{\prime}$ instead has one arc less than in the previous step. In fact, it has $a-1$ arcs, connecting $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , all oriented from $F_{1}^{\prime}$ to $F_{1}^{\prime\prime}$ and joining the vertex labelled $(a+1-(1+2k)r)^{\prime}$ of $F_{1}^{\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\prime}$ of $F_{1}^{\prime\prime}$ , for $k=1,\dotsc,a-1$ .
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<p data-bbox="110 386 500 428">Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\prime\prime}$ and $C_{2}^{\prime\prime}$ and gluing the curve $C_{2}^{\prime\prime}$ of the resulting disc with $C_{2}^{\prime}$ . </p>
<p data-bbox="109 429 500 602">The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\prime}$ and $E^{\prime\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ and they both have a unique vertex ( $w^{\prime}$ and $w^{\prime\prime}$ respectively). The cycle $E^{\prime}$ (resp. $E^{\prime\prime}$ ) is connected with $C_{1}^{\prime}$ (resp. with $C_{1}^{\prime\prime}$ ) by an arc joining $w^{\prime}$ (resp. $w^{\prime\prime}$ ) with the vertex labelled $(a+1)^{\prime}$ (resp. $(a+1-r)^{\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\prime}$ to $C_{1}^{\prime\prime}$ ; $2a+1$ of them belong to $D_{2}^{\prime}$ and the other $a$ belong to $D_{1}^{\prime}$ . More precisely, for each $i=1,\dots,2a+1$ , there is an arc of $D_{2}^{\prime}$ joining the vertex labelled $i$ of $C_{1}^{\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\prime\prime}$ ; while, for each $k=0,\dotsc,a-1$ , there is an arc of $D_{1}^{\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\prime}$ of $C_{1}^{\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\prime}$ of $C_{1}^{\prime\prime}$ . </p>
<p data-bbox="110 603 500 645">Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $C_{1}^{\prime\prime}$ of the resulting disc with $C_{1}^{\prime}$ . </p>
<p data-bbox="111 646 500 675">The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , which are copies of the cutting cycle $F_{1}$ . These cycles replace $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ and they both have one vertex less. It is easy to see that the cycle $D_{2}^{\prime}$ has exactly the same $2a+1$ arcs connecting $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , all oriented from $F_{1}^{\prime}$ to $F_{1}^{\prime\prime}$ ; if the labelling of the vertices of $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ is induced by the labelling of $F_{1}$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $D_{1}^{\prime}$ instead has one arc less than in the previous step. In fact, it has $a-1$ arcs, connecting $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$ , all oriented from $F_{1}^{\prime}$ to $F_{1}^{\prime\prime}$ and joining the vertex labelled $(a+1-(1+2k)r)^{\prime}$ of $F_{1}^{\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\prime}$ of $F_{1}^{\prime\prime}$ , for $k=1,\dotsc,a-1$ . </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 14}, {"type": "text", "text": "Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\\prime\\prime}$ and $C_{2}^{\\prime\\prime}$ and gluing the curve $C_{2}^{\\prime\\prime}$ of the resulting disc with $C_{2}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\\prime}$ and $E^{\\prime\\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ and they both have a unique vertex ( $w^{\\prime}$ and $w^{\\prime\\prime}$ respectively). The cycle $E^{\\prime}$ (resp. $E^{\\prime\\prime}$ ) is connected with $C_{1}^{\\prime}$ (resp. with $C_{1}^{\\prime\\prime}$ ) by an arc joining $w^{\\prime}$ (resp. $w^{\\prime\\prime}$ ) with the vertex labelled $(a+1)^{\\prime}$ (resp. $(a+1-r)^{\\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\\prime}$ to $C_{1}^{\\prime\\prime}$ ; $2a+1$ of them belong to $D_{2}^{\\prime}$ and the other $a$ belong to $D_{1}^{\\prime}$ . More precisely, for each $i=1,\\dots,2a+1$ , there is an arc of $D_{2}^{\\prime}$ joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\\prime\\prime}$ ; while, for each $k=0,\\dotsc,a-1$ , there is an arc of $D_{1}^{\\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $C_{1}^{\\prime\\prime}$ of the resulting disc with $C_{1}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , which are copies of the cutting cycle $F_{1}$ . These cycles replace $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ and they both have one vertex less. It is easy to see that the cycle $D_{2}^{\\prime}$ has exactly the same $2a+1$ arcs connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ ; if the labelling of the vertices of $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ is induced by the labelling of $F_{1}$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $D_{1}^{\\prime}$ instead has one arc less than in the previous step. In fact, it has $a-1$ arcs, connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ and joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $F_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $F_{1}^{\\prime\\prime}$ , for $k=1,\\dotsc,a-1$ . ", "page_idx": 14}]
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The cycles ", "type": "text"}, {"bbox": [371, 505, 384, 516], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [384, 504, 411, 517], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [412, 505, 426, 516], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [426, 504, 499, 517], "score": 1.0, "content": " are joined by", "type": "text"}], "index": 26}, {"bbox": [110, 518, 501, 532], "spans": [{"bbox": [110, 520, 141, 529], "score": 0.91, "content": "3a+1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [142, 518, 260, 532], "score": 1.0, "content": " arcs, all oriented from ", "type": "text"}, {"bbox": [261, 519, 273, 531], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [274, 518, 291, 532], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [291, 519, 306, 531], "score": 0.91, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [307, 518, 312, 532], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [313, 520, 344, 529], "score": 0.91, "content": "2a+1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [344, 518, 442, 532], "score": 1.0, "content": " of them belong to ", "type": "text"}, {"bbox": [442, 519, 456, 531], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [457, 518, 501, 532], "score": 1.0, "content": " and the", "type": "text"}], "index": 27}, {"bbox": [109, 532, 501, 547], "spans": [{"bbox": [109, 532, 140, 547], "score": 1.0, "content": "other ", "type": "text"}, {"bbox": [141, 537, 147, 542], "score": 0.87, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [147, 532, 202, 547], "score": 1.0, "content": " belong to ", "type": "text"}, {"bbox": [203, 533, 217, 545], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [218, 532, 350, 547], "score": 1.0, "content": ". More precisely, for each ", "type": "text"}, {"bbox": [351, 534, 438, 545], "score": 0.93, "content": "i=1,\\dots,2a+1", "type": "inline_equation", "height": 11, "width": 87}, {"bbox": [438, 532, 501, 547], "score": 1.0, "content": ", there is an", "type": "text"}], "index": 28}, {"bbox": [109, 546, 502, 561], "spans": [{"bbox": [109, 546, 142, 561], "score": 1.0, "content": "arc of ", "type": "text"}, {"bbox": [142, 548, 156, 560], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [157, 546, 296, 561], "score": 1.0, "content": " joining the vertex labelled ", "type": "text"}, {"bbox": [296, 549, 300, 557], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [300, 546, 316, 561], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [317, 548, 330, 560], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [330, 546, 457, 561], "score": 1.0, "content": " with the vertex labelled ", "type": "text"}, {"bbox": [457, 549, 486, 558], "score": 0.92, "content": "i-2r", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [486, 546, 502, 561], "score": 1.0, "content": " of", "type": "text"}], "index": 29}, {"bbox": [110, 560, 501, 576], "spans": [{"bbox": [110, 562, 125, 574], "score": 0.91, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [125, 560, 209, 576], "score": 1.0, "content": "; while, for each ", "type": "text"}, {"bbox": [209, 563, 293, 574], "score": 0.92, "content": "k=0,\\dotsc,a-1", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [293, 560, 390, 576], "score": 1.0, "content": ", there is an arc of ", "type": "text"}, {"bbox": [390, 562, 404, 574], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [405, 560, 501, 576], "score": 1.0, "content": " joining the vertex", "type": "text"}], "index": 30}, {"bbox": [110, 576, 499, 589], "spans": [{"bbox": [110, 576, 153, 589], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 577, 248, 589], "score": 0.91, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [248, 576, 263, 589], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [264, 577, 277, 589], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [277, 576, 404, 589], "score": 1.0, "content": " with the vertex labelled ", "type": "text"}, {"bbox": [405, 576, 499, 589], "score": 0.89, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 94}], "index": 31}, {"bbox": [108, 588, 143, 605], "spans": [{"bbox": [108, 588, 123, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 591, 138, 603], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [138, 588, 143, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 26.5, "bbox_fs": [108, 430, 502, 605]}, {"type": "text", "bbox": [110, 603, 500, 645], "lines": [{"bbox": [128, 604, 500, 619], "spans": [{"bbox": [128, 604, 446, 619], "score": 1.0, "content": "Apply again a Singer move of type IC, cutting along the cycle ", "type": "text"}, {"bbox": [446, 606, 459, 617], "score": 0.9, "content": "F_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [460, 604, 500, 619], "score": 1.0, "content": " (drawn", "type": "text"}], "index": 33}, {"bbox": [108, 617, 501, 636], "spans": [{"bbox": [108, 617, 232, 636], "score": 1.0, "content": "in Figure 5) containing ", "type": "text"}, {"bbox": [232, 620, 247, 632], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [248, 617, 273, 636], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [274, 620, 289, 629], "score": 0.92, "content": "E^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [289, 617, 402, 636], "score": 1.0, "content": " and gluing the curve ", "type": "text"}, {"bbox": [402, 620, 417, 632], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [417, 617, 501, 636], "score": 1.0, "content": " of the resulting", "type": "text"}], "index": 34}, {"bbox": [109, 630, 178, 649], "spans": [{"bbox": [109, 630, 160, 649], "score": 1.0, "content": "disc with ", "type": "text"}, {"bbox": [160, 635, 173, 646], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [174, 630, 178, 649], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "bbox_fs": [108, 604, 501, 649]}, {"type": "text", "bbox": [111, 646, 500, 675], "lines": [{"bbox": [127, 647, 499, 662], "spans": [{"bbox": [127, 647, 499, 662], "score": 1.0, "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the", "type": "text"}], "index": 36}, {"bbox": [110, 662, 499, 677], "spans": [{"bbox": [110, 662, 167, 677], "score": 1.0, "content": "new cycles ", "type": "text"}, {"bbox": [167, 663, 180, 676], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [180, 662, 205, 677], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 663, 220, 676], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [220, 662, 415, 677], "score": 1.0, "content": ", which are copies of the cutting cycle ", "type": "text"}, {"bbox": [415, 664, 427, 675], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [428, 662, 499, 677], "score": 1.0, "content": ". These cycles", "type": "text"}], "index": 37}, {"bbox": [110, 127, 501, 142], "spans": [{"bbox": [110, 127, 151, 142], "score": 1.0, "content": "replace ", "type": "text", "cross_page": true}, {"bbox": [151, 129, 164, 141], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [165, 127, 194, 142], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [194, 129, 208, 141], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [209, 127, 501, 142], "score": 1.0, "content": " and they both have one vertex less. It is easy to see", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 141, 501, 157], "spans": [{"bbox": [109, 141, 183, 157], "score": 1.0, "content": "that the cycle ", "type": "text", "cross_page": true}, {"bbox": [183, 144, 198, 155], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [198, 141, 310, 157], "score": 1.0, "content": " has exactly the same ", "type": "text", "cross_page": true}, {"bbox": [311, 144, 342, 154], "score": 0.93, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 31, "cross_page": true}, {"bbox": [342, 141, 426, 157], "score": 1.0, "content": " arcs connecting ", "type": "text", "cross_page": true}, {"bbox": [427, 144, 439, 155], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12, "cross_page": true}, {"bbox": [440, 141, 465, 157], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [465, 144, 480, 155], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [480, 141, 501, 157], "score": 1.0, "content": ", all", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 500, 170], "spans": [{"bbox": [110, 157, 181, 170], "score": 1.0, "content": "oriented from ", "type": "text", "cross_page": true}, {"bbox": [181, 158, 194, 170], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [194, 157, 210, 170], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [210, 158, 225, 170], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [225, 157, 393, 170], "score": 1.0, "content": "; if the labelling of the vertices of ", "type": "text", "cross_page": true}, {"bbox": [393, 158, 406, 170], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [406, 157, 430, 170], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [430, 158, 445, 170], "score": 0.93, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [445, 157, 500, 170], "score": 1.0, "content": " is induced", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [110, 171, 500, 185], "spans": [{"bbox": [110, 171, 204, 185], "score": 1.0, "content": "by the labelling of ", "type": "text", "cross_page": true}, {"bbox": [205, 173, 217, 183], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 10, "width": 12, "cross_page": true}, {"bbox": [217, 171, 500, 185], "score": 1.0, "content": " shown in Figure 5, these arcs join pairs of vertices with", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 363, 199], "score": 1.0, "content": "the same labelling of the previous step. The cycle ", "type": "text", "cross_page": true}, {"bbox": [364, 187, 378, 199], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [378, 185, 500, 199], "score": 1.0, "content": " instead has one arc less", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 199, 499, 214], "spans": [{"bbox": [109, 199, 324, 214], "score": 1.0, "content": "than in the previous step. In fact, it has ", "type": "text", "cross_page": true}, {"bbox": [324, 202, 351, 211], "score": 0.92, "content": "a-1", "type": "inline_equation", "height": 9, "width": 27, "cross_page": true}, {"bbox": [352, 199, 441, 214], "score": 1.0, "content": " arcs, connecting ", "type": "text", "cross_page": true}, {"bbox": [442, 201, 454, 213], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [455, 199, 481, 214], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [481, 201, 496, 213], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [496, 199, 499, 214], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [109, 214, 499, 228], "spans": [{"bbox": [109, 214, 198, 228], "score": 1.0, "content": "all oriented from ", "type": "text", "cross_page": true}, {"bbox": [198, 216, 210, 228], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [211, 214, 227, 228], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [228, 216, 242, 228], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [242, 214, 403, 228], "score": 1.0, "content": " and joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [404, 215, 499, 228], "score": 0.91, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 95, "cross_page": true}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 123, 243], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 230, 136, 242], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [136, 228, 263, 243], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [263, 230, 359, 243], "score": 0.93, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 96, "cross_page": true}, {"bbox": [360, 228, 375, 243], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [375, 230, 390, 242], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [390, 228, 414, 243], "score": 1.0, "content": ", for ", "type": "text", "cross_page": true}, {"bbox": [414, 231, 496, 242], "score": 0.93, "content": "k=1,\\dotsc,a-1", "type": "inline_equation", "height": 11, "width": 82, "cross_page": true}, {"bbox": [496, 228, 500, 243], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 7}], "index": 36.5, "bbox_fs": [110, 647, 499, 677]}]}
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[{"type": "text", "bbox": [109, 125, 501, 385], "content": "", "index": 0}, {"type": "text", "bbox": [110, 386, 500, 428], "content": "Now, apply to the diagram a Singer move of type IC, cutting along the cycle (drawn in Figure 4) containing and and gluing the curve of the resulting disc with .", "index": 1}, {"type": "text", "bbox": [109, 429, 500, 602], "content": "The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have a unique vertex ( and respectively). The cycle (resp. ) is connected with (resp. with ) by an arc joining (resp. ) with the vertex labelled (resp. , oriented as in Figure 5. The cycles and are joined by arcs, all oriented from to ; of them belong to and the other belong to . More precisely, for each , there is an arc of joining the vertex labelled of with the vertex labelled of ; while, for each , there is an arc of joining the vertex labelled of with the vertex labelled of .", "index": 2}, {"type": "text", "bbox": [110, 603, 500, 645], "content": "Apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 5) containing and and gluing the curve of the resulting disc with .", "index": 3}, {"type": "text", "bbox": [111, 646, 500, 675], "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have one vertex less. It is easy to see that the cycle has exactly the same arcs connecting and , all oriented from to ; if the labelling of the vertices of and is induced by the labelling of shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle instead has one arc less than in the previous step. In fact, it has arcs, connecting and , all oriented from to and joining the vertex labelled of with the vertex labelled of , for .", "index": 4}]
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[{"bbox": [126, 387, 500, 403], "content": "Now, apply to the diagram a Singer move of type IC, cutting along the", "parent_index": 1, "line_index": 0}, {"bbox": [110, 402, 498, 417], "content": "cycle (drawn in Figure 4) containing and and gluing the curve", "parent_index": 1, "line_index": 1}, {"bbox": [108, 413, 260, 433], "content": "of the resulting disc with .", "parent_index": 1, "line_index": 2}, {"bbox": [127, 430, 500, 445], "content": "The new Heegaard diagram obtained in this way shown in Figure 5. It", "parent_index": 2, "line_index": 0}, {"bbox": [110, 446, 500, 459], "content": "contains the new cycles and , which are copies of the cutting cycle .", "parent_index": 2, "line_index": 1}, {"bbox": [109, 459, 501, 476], "content": "These cycles replace and and they both have a unique vertex ( and", "parent_index": 2, "line_index": 2}, {"bbox": [110, 474, 501, 489], "content": "respectively). The cycle (resp. ) is connected with (resp. with", "parent_index": 2, "line_index": 3}, {"bbox": [110, 488, 500, 504], "content": ") by an arc joining (resp. ) with the vertex labelled (resp.", "parent_index": 2, "line_index": 4}, {"bbox": [110, 504, 499, 517], "content": ", oriented as in Figure 5. The cycles and are joined by", "parent_index": 2, "line_index": 5}, {"bbox": [110, 518, 501, 532], "content": "arcs, all oriented from to ; of them belong to and the", "parent_index": 2, "line_index": 6}, {"bbox": [109, 532, 501, 547], "content": "other belong to . More precisely, for each , there is an", "parent_index": 2, "line_index": 7}, {"bbox": [109, 546, 502, 561], "content": "arc of joining the vertex labelled of with the vertex labelled of", "parent_index": 2, "line_index": 8}, {"bbox": [110, 560, 501, 576], "content": "; while, for each , there is an arc of joining the vertex", "parent_index": 2, "line_index": 9}, {"bbox": [110, 576, 499, 589], "content": "labelled of with the vertex labelled", "parent_index": 2, "line_index": 10}, {"bbox": [108, 588, 143, 605], "content": "of .", "parent_index": 2, "line_index": 11}, {"bbox": [128, 604, 500, 619], "content": "Apply again a Singer move of type IC, cutting along the cycle (drawn", "parent_index": 3, "line_index": 0}, {"bbox": [108, 617, 501, 636], "content": "in Figure 5) containing and and gluing the curve of the resulting", "parent_index": 3, "line_index": 1}, {"bbox": [109, 630, 178, 649], "content": "disc with .", "parent_index": 3, "line_index": 2}, {"bbox": [127, 647, 499, 662], "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the", "parent_index": 4, "line_index": 0}, {"bbox": [110, 662, 499, 677], "content": "new cycles and , which are copies of the cutting cycle . These cycles", "parent_index": 4, "line_index": 1}, {"bbox": [110, 127, 501, 142], "content": "replace and and they both have one vertex less. It is easy to see", "parent_index": 4, "line_index": 2}, {"bbox": [109, 141, 501, 157], "content": "that the cycle has exactly the same arcs connecting and , all", "parent_index": 4, "line_index": 3}, {"bbox": [110, 157, 500, 170], "content": "oriented from to ; if the labelling of the vertices of and is induced", "parent_index": 4, "line_index": 4}, {"bbox": [110, 171, 500, 185], "content": "by the labelling of shown in Figure 5, these arcs join pairs of vertices with", "parent_index": 4, "line_index": 5}, {"bbox": [109, 185, 500, 199], "content": "the same labelling of the previous step. The cycle instead has one arc less", "parent_index": 4, "line_index": 6}, {"bbox": [109, 199, 499, 214], "content": "than in the previous step. In fact, it has arcs, connecting and ,", "parent_index": 4, "line_index": 7}, {"bbox": [109, 214, 499, 228], "content": "all oriented from to and joining the vertex labelled", "parent_index": 4, "line_index": 8}, {"bbox": [109, 228, 500, 243], "content": "of with the vertex labelled of , for .", "parent_index": 4, "line_index": 9}]
|
[]
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[{"bbox": [139, 404, 149, 413], "content": "E", "parent_index": 1, "subtype": "inline"}, {"bbox": [315, 403, 329, 415], "content": "C_{1}^{\\prime\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [356, 403, 371, 415], "content": "C_{2}^{\\prime\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [484, 403, 498, 415], "content": "C_{2}^{\\prime\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [242, 418, 255, 430], "content": "C_{2}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [234, 447, 246, 456], "content": "E^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [273, 447, 288, 456], "content": "E^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [486, 447, 496, 456], "content": "E", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 461, 230, 473], "content": "C_{2}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [256, 461, 271, 473], "content": "C_{2}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [465, 461, 476, 471], "content": "w^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 476, 124, 485], "content": "w^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [255, 476, 268, 485], "content": "E^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [307, 476, 321, 485], "content": "E^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [424, 476, 437, 488], "content": "C_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 490, 124, 502], "content": "C_{1}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [226, 490, 237, 499], "content": "w^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [277, 490, 291, 500], "content": "w^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [427, 490, 466, 502], "content": "(a+1)^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 504, 174, 517], "content": "(a+1-r)^{\\prime})", "parent_index": 2, "subtype": "inline"}, {"bbox": [371, 505, 384, 516], "content": "C_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [412, 505, 426, 516], "content": "C_{1}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 520, 141, 529], "content": "3a+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [261, 519, 273, 531], "content": "C_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [291, 519, 306, 531], "content": "C_{1}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [313, 520, 344, 529], "content": "2a+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [442, 519, 456, 531], "content": "D_{2}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [141, 537, 147, 542], "content": "a", "parent_index": 2, "subtype": "inline"}, {"bbox": [203, 533, 217, 545], "content": "D_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [351, 534, 438, 545], "content": "i=1,\\dots,2a+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [142, 548, 156, 560], "content": "D_{2}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [296, 549, 300, 557], "content": "i", "parent_index": 2, "subtype": "inline"}, {"bbox": [317, 548, 330, 560], "content": "C_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [457, 549, 486, 558], "content": "i-2r", "parent_index": 2, "subtype": "inline"}, {"bbox": [110, 562, 125, 574], "content": "C_{1}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [209, 563, 293, 574], "content": "k=0,\\dotsc,a-1", "parent_index": 2, "subtype": "inline"}, {"bbox": [390, 562, 404, 574], "content": "D_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [153, 577, 248, 589], "content": "(a+1-(1+2k)r)^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [264, 577, 277, 589], "content": "C_{1}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 576, 499, 589], "content": "(a+1-(3+2k)r)^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [123, 591, 138, 603], "content": "C_{1}^{\\prime\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [446, 606, 459, 617], "content": "F_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [232, 620, 247, 632], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [274, 620, 289, 629], "content": "E^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [402, 620, 417, 632], "content": "C_{1}^{\\prime\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [160, 635, 173, 646], "content": "C_{1}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [167, 663, 180, 676], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [205, 663, 220, 676], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [415, 664, 427, 675], "content": "F_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [151, 129, 164, 141], "content": "C_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [194, 129, 208, 141], "content": "C_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [183, 144, 198, 155], "content": "D_{2}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [311, 144, 342, 154], "content": "2a+1", "parent_index": 4, "subtype": "inline"}, {"bbox": [427, 144, 439, 155], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [465, 144, 480, 155], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [181, 158, 194, 170], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [210, 158, 225, 170], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [393, 158, 406, 170], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [430, 158, 445, 170], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [205, 173, 217, 183], "content": "F_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 187, 378, 199], "content": "D_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [324, 202, 351, 211], "content": "a-1", "parent_index": 4, "subtype": "inline"}, {"bbox": [442, 201, 454, 213], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [481, 201, 496, 213], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [198, 216, 210, 228], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 216, 242, 228], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [404, 215, 499, 228], "content": "(a+1-(1+2k)r)^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 230, 136, 242], "content": "F_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [263, 230, 359, 243], "content": "(a+1-(3+2k)r)^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [375, 230, 390, 242], "content": "F_{1}^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [414, 231, 496, 242], "content": "k=1,\\dotsc,a-1", "parent_index": 4, "subtype": "inline"}]
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[]
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Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{1}^{\prime\prime}$ of the resulting disc with $F_{1}^{\prime}$ .
The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{h-1}^{\prime\prime}$ of the resulting disc with $F_{h-1}^{\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\prime}$ contains only two arcs, none of which connects $F_{a}^{\prime}$ with $F_{a}^{\prime\prime}$ .
After the move of type IC corresponding to $h\,=\,a\,+\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\prime},E^{\prime\prime}$ and by the cycle $D_{1}^{\prime}$ , composed by a unique arc connecting $E^{\prime}$ with $E^{\prime\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ .
# References
[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $5_{2}$ . Proc. Edinb. Math. Soc. 42 (1999), 575–587
[2] Berge, J.: The knots in $D^{2}\times S^{1}$ with non-trivial Dehn surgery yielding $D^{2}\times S^{1}$ . Topology Appl. 38 (1991), 1–19
[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sieradsky groups. Algebra Colloq. 5 (1998), 203–217
[4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the minisemester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56
[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717
[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55
[7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6
[8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235
[9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179
[10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65
[11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206
[12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23
[13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976
[14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96
[15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980
[16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235
[17] Kim, G., Kim, Y., Vesnin, A.: The knot $5_{2}$ and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980
[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000
[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284
[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Transform. Groups 2 (1997), 165–182
[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68
[22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123
[23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167
[24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263
[25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613
[26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471
[27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243
[28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251
[29] Reni, M., Zimmermann, B.,: Extending finite group actions from surfaces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887
[30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170
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<html><body>
<p data-bbox="110 241 500 284">Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{1}^{\prime\prime}$ of the resulting disc with $F_{1}^{\prime}$ . </p>
<p data-bbox="110 285 500 371">The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{h-1}^{\prime\prime}$ of the resulting disc with $F_{h-1}^{\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\prime}$ contains only two arcs, none of which connects $F_{a}^{\prime}$ with $F_{a}^{\prime\prime}$ . </p>
<p data-bbox="110 372 500 472">After the move of type IC corresponding to $h\,=\,a\,+\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\prime},E^{\prime\prime}$ and by the cycle $D_{1}^{\prime}$ , composed by a unique arc connecting $E^{\prime}$ with $E^{\prime\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ . </p>
<h1 data-bbox="109 493 201 512">References </h1>
<p data-bbox="109 523 501 670">[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $5_{2}$ . Proc. Edinb. Math. Soc. 42 (1999), 575–587
[2] Berge, J.: The knots in $D^{2}\times S^{1}$ with non-trivial Dehn surgery yielding $D^{2}\times S^{1}$ . Topology Appl. 38 (1991), 1–19
[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sieradsky groups. Algebra Colloq. 5 (1998), 203–217
[4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the minisemester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56
[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717
[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55
[7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6
[8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235
[9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179
[10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65
[11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206
[12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23
[13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976
[14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96
[15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980
[16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235
[17] Kim, G., Kim, Y., Vesnin, A.: The knot $5_{2}$ and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980
[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000
[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284
[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Transform. Groups 2 (1997), 165–182
[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68
[22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123
[23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167
[24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263
[25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613
[26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471
[27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243
[28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251
[29] Reni, M., Zimmermann, B.,: Extending finite group actions from surfaces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887
[30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170 </p>
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0003042v1
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[{"type": "text", "text": "", "page_idx": 15}, {"type": "text", "text": "Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{1}^{\\prime\\prime}$ of the resulting disc with $F_{1}^{\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{h-1}^{\\prime\\prime}$ of the resulting disc with $F_{h-1}^{\\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\\prime}$ contains only two arcs, none of which connects $F_{a}^{\\prime}$ with $F_{a}^{\\prime\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "After the move of type IC corresponding to $h\\,=\\,a\\,+\\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\\prime},E^{\\prime\\prime}$ and by the cycle $D_{1}^{\\prime}$ , composed by a unique arc connecting $E^{\\prime}$ with $E^{\\prime\\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ . ", "page_idx": 15}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $5_{2}$ . Proc. Edinb. Math. Soc. 42 (1999), 575–587 \n[2] Berge, J.: The knots in $D^{2}\\times S^{1}$ with non-trivial Dehn surgery yielding $D^{2}\\times S^{1}$ . 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It is easy to see", "type": "text"}], "index": 0}, {"bbox": [109, 141, 501, 157], "spans": [{"bbox": [109, 141, 183, 157], "score": 1.0, "content": "that the cycle ", "type": "text"}, {"bbox": [183, 144, 198, 155], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [198, 141, 310, 157], "score": 1.0, "content": " has exactly the same ", "type": "text"}, {"bbox": [311, 144, 342, 154], "score": 0.93, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [342, 141, 426, 157], "score": 1.0, "content": " arcs connecting ", "type": "text"}, {"bbox": [427, 144, 439, 155], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [440, 141, 465, 157], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [465, 144, 480, 155], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [480, 141, 501, 157], "score": 1.0, "content": ", all", "type": "text"}], "index": 1}, {"bbox": [110, 157, 500, 170], "spans": [{"bbox": [110, 157, 181, 170], "score": 1.0, "content": "oriented from ", "type": "text"}, {"bbox": [181, 158, 194, 170], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [194, 157, 210, 170], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [210, 158, 225, 170], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [225, 157, 393, 170], "score": 1.0, "content": "; if the labelling of the vertices of ", "type": "text"}, {"bbox": [393, 158, 406, 170], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [406, 157, 430, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [430, 158, 445, 170], "score": 0.93, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [445, 157, 500, 170], "score": 1.0, "content": " is induced", "type": "text"}], "index": 2}, {"bbox": [110, 171, 500, 185], "spans": [{"bbox": [110, 171, 204, 185], "score": 1.0, "content": "by the labelling of ", "type": "text"}, {"bbox": [205, 173, 217, 183], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [217, 171, 500, 185], "score": 1.0, "content": " shown in Figure 5, these arcs join pairs of vertices with", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 363, 199], "score": 1.0, "content": "the same labelling of the previous step. 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Math.", "type": "text", "cross_page": true}], "index": 27, "is_list_start_line": true}, {"bbox": [127, 651, 236, 668], "spans": [{"bbox": [127, 651, 236, 668], "score": 1.0, "content": "184 (1995), 231–235", "type": "text", "cross_page": true}], "index": 28, "is_list_end_line": true}, {"bbox": [110, 128, 500, 143], "spans": [{"bbox": [110, 128, 358, 143], "score": 1.0, "content": "[17] Kim, G., Kim, Y., Vesnin, A.: The knot ", "type": "text", "cross_page": true}, {"bbox": [358, 130, 369, 140], "score": 0.75, "content": "5_{2}", "type": "inline_equation", "height": 10, "width": 11, "cross_page": true}, {"bbox": [369, 128, 500, 143], "score": 1.0, "content": " and cyclically presented", "type": "text", "cross_page": true}], "index": 0, "is_list_start_line": true}, {"bbox": [127, 142, 385, 158], "spans": [{"bbox": [127, 142, 385, 158], "score": 1.0, "content": "groups. J. Korean Math. Soc. 35 (1998), 961–980", "type": "text", "cross_page": true}], "index": 1, "is_list_end_line": true}, {"bbox": [109, 166, 501, 183], "spans": [{"bbox": [109, 166, 501, 183], "score": 1.0, "content": "[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To", "type": "text", "cross_page": true}], "index": 2, "is_list_start_line": true}, {"bbox": [127, 181, 500, 196], "spans": [{"bbox": [127, 181, 500, 196], "score": 1.0, "content": "appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter,", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [128, 195, 247, 210], "spans": [{"bbox": [128, 195, 247, 210], "score": 1.0, "content": "Berlin-New York, 2000", "type": "text", "cross_page": true}], "index": 4, "is_list_end_line": true}, {"bbox": [110, 220, 501, 235], "spans": [{"bbox": [110, 220, 501, 235], "score": 1.0, "content": "[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57", "type": "text", "cross_page": true}], "index": 5, "is_list_start_line": true}, {"bbox": [128, 235, 212, 250], "spans": [{"bbox": [128, 235, 212, 250], "score": 1.0, "content": "(1985), 261–284", "type": "text", "cross_page": true}], "index": 6, "is_list_end_line": true}, {"bbox": [110, 259, 500, 274], "spans": [{"bbox": [110, 259, 500, 274], "score": 1.0, "content": "[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans-", "type": "text", "cross_page": true}], "index": 7, "is_list_start_line": true}, {"bbox": [127, 272, 295, 289], "spans": [{"bbox": [127, 272, 295, 289], "score": 1.0, "content": "form. Groups 2 (1997), 165–182", "type": "text", "cross_page": true}], "index": 8, "is_list_end_line": true}, {"bbox": [110, 297, 499, 313], "spans": [{"bbox": [110, 297, 499, 313], "score": 1.0, "content": "[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links.", "type": "text", "cross_page": true}], "index": 9, "is_list_start_line": true}, {"bbox": [128, 312, 371, 327], "spans": [{"bbox": [128, 312, 371, 327], "score": 1.0, "content": "Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68", "type": "text", "cross_page": true}], "index": 10, "is_list_end_line": true}, {"bbox": [109, 335, 501, 353], "spans": [{"bbox": [109, 335, 501, 353], "score": 1.0, "content": "[22] Montesinos, J.M.: Representing 3-manifolds by a universal branching", "type": "text", "cross_page": true}], "index": 11, "is_list_start_line": true}, {"bbox": [127, 351, 416, 366], "spans": [{"bbox": [127, 351, 416, 366], "score": 1.0, "content": "set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123", "type": "text", "cross_page": true}], "index": 12, "is_list_end_line": true}, {"bbox": [110, 375, 499, 390], "spans": [{"bbox": [110, 375, 499, 390], "score": 1.0, "content": "[23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math.", "type": "text", "cross_page": true}], "index": 13, "is_list_start_line": true}, {"bbox": [127, 389, 264, 405], "spans": [{"bbox": [127, 389, 264, 405], "score": 1.0, "content": "Ann. 289 (1991), 143–167", "type": "text", "cross_page": true}], "index": 14, "is_list_end_line": true}, {"bbox": [109, 412, 502, 430], "spans": [{"bbox": [109, 412, 502, 430], "score": 1.0, "content": "[24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of", "type": "text", "cross_page": true}], "index": 15, "is_list_start_line": true}, {"bbox": [127, 428, 399, 444], "spans": [{"bbox": [127, 428, 399, 444], "score": 1.0, "content": "S . J. Knot Theory Ramifications 5 (1996), 239–263", "type": "text", "cross_page": true}], "index": 16, "is_list_end_line": true}, {"bbox": [110, 453, 500, 468], "spans": [{"bbox": [110, 453, 500, 468], "score": 1.0, "content": "[25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from", "type": "text", "cross_page": true}], "index": 17, "is_list_start_line": true}, {"bbox": [127, 467, 430, 482], "spans": [{"bbox": [127, 467, 430, 482], "score": 1.0, "content": "2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613", "type": "text", "cross_page": true}], "index": 18, "is_list_end_line": true}, {"bbox": [109, 490, 500, 508], "spans": [{"bbox": [109, 490, 500, 508], "score": 1.0, "content": "[26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "type": "text", "cross_page": true}], "index": 19, "is_list_start_line": true}, {"bbox": [126, 505, 434, 521], "spans": [{"bbox": [126, 505, 434, 521], "score": 1.0, "content": "spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471", "type": "text", "cross_page": true}], "index": 20, "is_list_end_line": true}, {"bbox": [109, 530, 501, 547], "spans": [{"bbox": [109, 530, 501, 547], "score": 1.0, "content": "[27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "type": "text", "cross_page": true}], "index": 21, "is_list_start_line": true}, {"bbox": [127, 545, 493, 560], "spans": [{"bbox": [127, 545, 493, 560], "score": 1.0, "content": "spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243", "type": "text", "cross_page": true}], "index": 22}, {"bbox": [109, 568, 500, 586], "spans": [{"bbox": [109, 568, 500, 586], "score": 1.0, "content": "[28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "type": "text", "cross_page": true}], "index": 23, "is_list_start_line": true}, {"bbox": [126, 583, 497, 599], "spans": [{"bbox": [126, 583, 497, 599], "score": 1.0, "content": "spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251", "type": "text", "cross_page": true}], "index": 24}, {"bbox": [109, 608, 500, 625], "spans": [{"bbox": [109, 608, 500, 625], "score": 1.0, "content": "[29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur-", "type": "text", "cross_page": true}], "index": 25, "is_list_start_line": true}, {"bbox": [126, 622, 479, 638], "spans": [{"bbox": [126, 622, 479, 638], "score": 1.0, "content": "faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887", "type": "text", "cross_page": true}], "index": 26, "is_list_end_line": true}, {"bbox": [110, 647, 493, 662], "spans": [{"bbox": [110, 647, 493, 662], "score": 0.9834961295127869, "content": "[30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170", "type": "text", "cross_page": true}], "index": 27, "is_list_start_line": true}], "index": 28.5, "bbox_fs": [109, 526, 500, 671]}]}
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[{"type": "text", "bbox": [110, 124, 500, 240], "content": "", "index": 0}, {"type": "text", "bbox": [110, 241, 500, 284], "content": "Now, apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 6) containing and and gluing the curve of the resulting disc with .", "index": 1}, {"type": "text", "bbox": [110, 285, 500, 371], "content": "The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle . By inductive application of Singer moves of type IC, cutting along the cycle (drawn in Figure 7) containing and and gluing the curve of the resulting disc with , we obtain, for , the situation shown in Figure 8, where the cycle contains only two arcs, none of which connects with .", "index": 2}, {"type": "text", "bbox": [110, 372, 500, 472], "content": "After the move of type IC corresponding to , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles and by the cycle , composed by a unique arc connecting with . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space .", "index": 3}, {"type": "title", "bbox": [109, 493, 201, 512], "content": "References", "index": 4}, {"type": "list", "bbox": [109, 523, 501, 670], "content": "", "index": 5}]
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[{"bbox": [126, 243, 499, 258], "content": "Now, apply again a Singer move of type IC, cutting along the cycle", "parent_index": 1, "line_index": 0}, {"bbox": [111, 257, 498, 271], "content": "(drawn in Figure 6) containing and and gluing the curve of the", "parent_index": 1, "line_index": 1}, {"bbox": [110, 272, 225, 286], "content": "resulting disc with .", "parent_index": 1, "line_index": 2}, {"bbox": [127, 286, 499, 300], "content": "The new Heegaard diagram only differs from the previous one for con-", "parent_index": 2, "line_index": 0}, {"bbox": [109, 301, 501, 315], "content": "taining one arc less in the cycle . By inductive application of Singer moves", "parent_index": 2, "line_index": 1}, {"bbox": [109, 315, 498, 332], "content": "of type IC, cutting along the cycle (drawn in Figure 7) containing", "parent_index": 2, "line_index": 2}, {"bbox": [108, 328, 501, 346], "content": "and and gluing the curve of the resulting disc with , we obtain,", "parent_index": 2, "line_index": 3}, {"bbox": [109, 344, 498, 358], "content": "for , the situation shown in Figure 8, where the cycle contains only", "parent_index": 2, "line_index": 4}, {"bbox": [109, 358, 343, 374], "content": "two arcs, none of which connects with .", "parent_index": 2, "line_index": 5}, {"bbox": [127, 372, 500, 388], "content": "After the move of type IC corresponding to , we obtain the", "parent_index": 3, "line_index": 0}, {"bbox": [109, 388, 500, 402], "content": "situation of Figure 9 in which the Heegaard diagram contains a pair of com-", "parent_index": 3, "line_index": 1}, {"bbox": [109, 402, 500, 417], "content": "plementary handles given by the pair of cycles and by the cycle ,", "parent_index": 3, "line_index": 2}, {"bbox": [109, 417, 502, 430], "content": "composed by a unique arc connecting with . The deletion of this pair of", "parent_index": 3, "line_index": 3}, {"bbox": [109, 431, 500, 445], "content": "complementary handles (Singer move of type III) leads to the genus one Hee-", "parent_index": 3, "line_index": 4}, {"bbox": [109, 446, 500, 460], "content": "gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram", "parent_index": 3, "line_index": 5}, {"bbox": [109, 460, 268, 474], "content": "of the lens space .", "parent_index": 3, "line_index": 6}, {"bbox": [110, 496, 202, 513], "content": "References", "parent_index": 4, "line_index": 0}, {"bbox": [110, 526, 500, 542], "content": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the", "parent_index": 5, "line_index": 0}, {"bbox": [128, 542, 401, 555], "content": "knot . Proc. Edinb. Math. Soc. 42 (1999), 575–587", "parent_index": 5, "line_index": 1}, {"bbox": [109, 563, 500, 580], "content": "[2] Berge, J.: The knots in with non-trivial Dehn surgery yielding", "parent_index": 5, "line_index": 2}, {"bbox": [128, 579, 343, 595], "content": ". Topology Appl. 38 (1991), 1–19", "parent_index": 5, "line_index": 3}, {"bbox": [110, 603, 500, 619], "content": "[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad-", "parent_index": 5, "line_index": 4}, {"bbox": [128, 618, 367, 633], "content": "sky groups. Algebra Colloq. 5 (1998), 203–217", "parent_index": 5, "line_index": 5}, {"bbox": [110, 642, 500, 657], "content": "[4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and", "parent_index": 5, "line_index": 6}, {"bbox": [128, 657, 500, 671], "content": "cyclic presentations of groups. In: Knot theory. Proceedings of the mini-", "parent_index": 5, "line_index": 7}, {"bbox": [126, 127, 501, 143], "content": "semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish", "parent_index": 5, "line_index": 8}, {"bbox": [128, 142, 501, 157], "content": "Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42", "parent_index": 5, "line_index": 9}, {"bbox": [128, 155, 199, 170], "content": "(1998), 49–56", "parent_index": 5, "line_index": 10}, {"bbox": [109, 181, 499, 196], "content": "[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann.", "parent_index": 5, "line_index": 11}, {"bbox": [128, 196, 236, 210], "content": "294 (1992), 701–717", "parent_index": 5, "line_index": 12}, {"bbox": [109, 219, 500, 236], "content": "[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. 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Note Ser., vol. 42, Cambridge Univ. Press, Cambridge,", "parent_index": 5, "line_index": 33}, {"bbox": [127, 612, 187, 629], "content": "U.K., 1980", "parent_index": 5, "line_index": 34}, {"bbox": [109, 636, 499, 653], "content": "[16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math.", "parent_index": 5, "line_index": 35}, {"bbox": [127, 651, 236, 668], "content": "184 (1995), 231–235", "parent_index": 5, "line_index": 36}, {"bbox": [110, 128, 500, 143], "content": "[17] Kim, G., Kim, Y., Vesnin, A.: The knot and cyclically presented", "parent_index": 5, "line_index": 37}, {"bbox": [127, 142, 385, 158], "content": "groups. J. Korean Math. Soc. 35 (1998), 961–980", "parent_index": 5, "line_index": 38}, {"bbox": [109, 166, 501, 183], "content": "[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To", "parent_index": 5, "line_index": 39}, {"bbox": [127, 181, 500, 196], "content": "appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter,", "parent_index": 5, "line_index": 40}, {"bbox": [128, 195, 247, 210], "content": "Berlin-New York, 2000", "parent_index": 5, "line_index": 41}, {"bbox": [110, 220, 501, 235], "content": "[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57", "parent_index": 5, "line_index": 42}, {"bbox": [128, 235, 212, 250], "content": "(1985), 261–284", "parent_index": 5, "line_index": 43}, {"bbox": [110, 259, 500, 274], "content": "[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans-", "parent_index": 5, "line_index": 44}, {"bbox": [127, 272, 295, 289], "content": "form. Groups 2 (1997), 165–182", "parent_index": 5, "line_index": 45}, {"bbox": [110, 297, 499, 313], "content": "[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links.", "parent_index": 5, "line_index": 46}, {"bbox": [128, 312, 371, 327], "content": "Mem. Am. Math. 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Knot Theory Ramifications 5 (1996), 239–263", "parent_index": 5, "line_index": 53}, {"bbox": [110, 453, 500, 468], "content": "[25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from", "parent_index": 5, "line_index": 54}, {"bbox": [127, 467, 430, 482], "content": "2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613", "parent_index": 5, "line_index": 55}, {"bbox": [109, 490, 500, 508], "content": "[26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "parent_index": 5, "line_index": 56}, {"bbox": [126, 505, 434, 521], "content": "spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471", "parent_index": 5, "line_index": 57}, {"bbox": [109, 530, 501, 547], "content": "[27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "parent_index": 5, "line_index": 58}, {"bbox": [127, 545, 493, 560], "content": "spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243", "parent_index": 5, "line_index": 59}, {"bbox": [109, 568, 500, 586], "content": "[28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to", "parent_index": 5, "line_index": 60}, {"bbox": [126, 583, 497, 599], "content": "spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251", "parent_index": 5, "line_index": 61}, {"bbox": [109, 608, 500, 625], "content": "[29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur-", "parent_index": 5, "line_index": 62}, {"bbox": [126, 622, 479, 638], "content": "faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887", "parent_index": 5, "line_index": 63}, {"bbox": [110, 647, 493, 662], "content": "[30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170", "parent_index": 5, "line_index": 64}]
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[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111
[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and $(1,1)$ -decomposible knots. Preprint, 1999
[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664
[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topology 31 (1992), 271–279
[36] Wu, Y-Q.,: $\Dot{O}$ -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331
[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696
[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610
LUIGI GRASSELLI, Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected]
MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected]
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<p data-bbox="110 124 501 154">[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111 </p>
<p data-bbox="110 163 501 193">[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and $(1,1)$ -decomposible knots. Preprint, 1999 </p>
<p data-bbox="110 241 501 271">[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664 </p>
<p data-bbox="110 279 501 310">[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topology 31 (1992), 271–279 </p>
<p data-bbox="110 318 501 349">[36] Wu, Y-Q.,: $\Dot{O}$ -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331 </p>
<p data-bbox="110 357 500 388">[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696 </p>
<p data-bbox="110 397 499 426">[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610 </p>
<p data-bbox="109 453 500 498">LUIGI GRASSELLI, Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected] </p>
<p data-bbox="109 511 501 556">MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected] </p>
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Figure 6:
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<div class="image" data-bbox="167 120 444 634"><img data-bbox="167 120 444 634"/><p class="caption" data-bbox="277 657 326 671">Figure 6: </p></div>
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0003042v1
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Figure 7:
Figure 8:
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<html><body>
<div class="image" data-bbox="195 129 414 335"><img data-bbox="195 129 414 335"/><p class="caption" data-bbox="277 357 326 372">Figure 7: </p></div>
<div class="image" data-bbox="192 391 419 624"><img data-bbox="192 391 419 624"/><p class="caption" data-bbox="277 650 326 664">Figure 8: </p></div>
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0003042v1
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Figure 9:
Figure 10:
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<html><body>
<div class="image" data-bbox="189 163 422 370"><img data-bbox="189 163 422 370"/><p class="caption" data-bbox="277 392 326 407">Figure 9: </p></div>
<div class="image" data-bbox="181 498 427 593"><img data-bbox="181 498 427 593"/><p class="caption" data-bbox="275 614 329 629">Figure 10: </p></div>
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0003042v1
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[{"type": "image", "bbox": [167, 120, 444, 634], "content": "", "index": 0}]
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[]
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[{"bbox": [167, 120, 444, 634], "content": "", "parent_index": 0, "subtype": "body"}, {"bbox": [277, 657, 326, 671], "content": "Figure 6:", "parent_index": 0, "subtype": "caption"}]
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[]
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[]
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