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<html>
<head>
<link rel="stylesheet" type="text/css" href="regina.css">
<title>Regina - Supporting Data</title>
</head>
<body bgcolor="#FFFFFF" text="#000000"
link="#0000EE" vlink="#551A8B" alink="#FF0000">
<h1><a name="contents">Regina - Supporting Data</a></h1>
<p>
<table class="contentswrapper" cellspacing=0>
<tbody>
<tr><td valign="top">
<table class="contents" cellspacing=0><tbody>
<tr><td><a href="#knots">Knot tables</a></td></tr>
<tr><td><a href="#census">3-manifold census data</a></td></tr>
<tr><td><a href="#weber-seifert">Weber-Seifert dodecahedral space</a></td></tr>
<tr><td><a href="index.html">Back to main page ...</a></td></tr>
</tbody></table>
</td></tr>
</tbody>
</table>
<h2><a name="knots">Knot tables</a></h2>
Regina now includes native
support for knots and links.
If you wish to play with some smaller examples, you can open Regina and
select <i>Open Example</i> → <i>Prime Knots</i> from the menu.
<p>
If you want more, then here you can
download the tables of all 352,152,252 prime non-trivial knots with up to
19 crossings.
<p>
The tables are plain text CSV (comma-separated value) files which you can
load into a spreadsheet and/or process with a text editor, and have been
compressed with <tt>bzip2</tt>. The fields include:
<p>
<ul>
<li><p><b>name:</b> The name of the knot, using a naming scheme
specific to these tables.
An example name is <tt>12nh_137</tt>.
In general the name is of the form <i>c[an][tsh]_k</i>, where:
<ul><li><i>c</i> is the number of crossings;</li>
<li><i>[an]</i> indicates whether the knot is alternating or
non-alternating;</li>
<li><i>[tsh]</i> indicates whether the knot is
a torus, satellite or hyperbolic knot;</li>
<li><i>k</i> is a positive integer that sorts the knots
within each of these classes.</li></ul>
<p>
The torus and satellite knots are sorted according to their
structure. The hyperbolic knots are sorted <i>roughly</i> by
volume, but take care—although the distinctness and hyperbolicity
of the knots are proven using exact computation, the final sorting order
is based on approximate volume computation only.
</li>
<li><p><b>knot_sig:</b> The knot diagram, expressed as a native Regina
knot signature. An example signature (for the knot <tt>7ah_5</tt>)
is <tt>habcadebcfgedgfvvb-Za</tt>.
<i>Knot signatures</i> uniquely identify a diagram on
the 2-sphere up to relabelling and/or reflection. In Regina 5.2 or
later you can reconstruct a knot from a signature through the GUI, or by
calling <tt>Link.fromKnotSig()</tt> in python.
<li><p><b>dt_code:</b> The knot diagram, expressed as an alphabetical
Dowker-Thistlethwaite (DT) code An example DT code (again for the knot
<tt>7ah_5</tt>) is <tt>fdgeacb</tt>.
<li><p><b>dt_name:</b> Identifies the knot in other online databases such as
<a href="http://www.indiana.edu/~knotinfo/">Knotinfo</a> and
<!--a href="http://pzacad.pitzer.edu/~jhoste/HosteWebPages/kntscp.html">Knotscape</a-->
<a href="http://www.math.utk.edu/~morwen/knotscape.html">Knotscape</a>
(this field only appears in the tables for
≤ 12 crossings). This field uses the
Dowker-Thistlethwaite naming convention, where knots are numbered
according to their minimal DT codes.
<li><p><b>structure:</b> Gives the full structure of a torus or
satellite knot (this field does not appear in the hyperbolic tables).
An example is <tt>Trefoil[-3/2]</tt>, indicating a
satellite formed by inserting the rational tangle -3/2 into the double
of the right-hand trefoil. See the paper below for a full explanation
of what the various structure descriptions mean.
</ul>
<p>
<b>Citation:</b> If you wish to cite this data, please reference:
<ul>
<li>Benjamin A. Burton,
<a href="https://drops.dagstuhl.de/opus/volltexte/2020/12183/"><i>The next 350 million knots</i></a>,
36th International Symposium on Computational Geometry (SoCG 2020)
(S. Cabello, D.Z. Chen, eds.),
Leibniz Int. Proc. Inform., vol. 164,
Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2020,
pp. 25:1–25:17.</li>
</ul>
<p>
<a href="http://www.maths.uq.edu.au/~bab/knots/all_3-12.tar.bz2">Download all 3–12 crossing knots at once</a> (56 kB)<br>
<a href="http://www.maths.uq.edu.au/~bab/knots/all_13-16.tar.bz2">Download all 13–16 crossing knots at once</a> (41 MB)<br>
Download individual tables (up to 19 crossings) below:
<p>
<table cellspacing=0 border=0 class="data"><tbody>
<tr>
<th class="first" colspan=2>Crossings</th>
<th>Torus</th>
<th>Satellite</th>
<th colspan=2>Hyperbolic</th>
</tr>
<tr>
<td class="knotcross">3</td><td>alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/3a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall">—</td>
<td class="knotcount">—</td><td> </td>
</tr>
<tr>
<td class="knotcross">4</td><td>alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/4a-hyp.csv.bz2">1 knot</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross">5</td><td>alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/5a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/5a-hyp.csv.bz2">1 knot</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross">6</td><td>alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/6a-hyp.csv.bz2">3 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross">7</td><td>alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/7a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/7a-hyp.csv.bz2">6 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>8</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/8a-hyp.csv.bz2">18 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/8n-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/8n-hyp.csv.bz2">2 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>9</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/9a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/9a-hyp.csv.bz2">40 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/9n-hyp.csv.bz2">8 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>10</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/10a-hyp.csv.bz2">123 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/10n-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/10n-hyp.csv.bz2">41 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>11</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/11a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/11a-hyp.csv.bz2">366 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/11n-hyp.csv.bz2">185 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>12</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/12a-hyp.csv.bz2">1,288 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall">—</td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/12n-hyp.csv.bz2">888 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>13</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/13a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/13a-hyp.csv.bz2">4,877 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/13n-satellite.csv.bz2">2 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/13n-hyp.csv.bz2">5,108 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>14</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/14a-hyp.csv.bz2">19,536 knots</a></td><td class="knotsizeupper"> </td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/14n-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/14n-satellite.csv.bz2">2 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/14n-hyp.csv.bz2">27,433 knots</a></td><td class="knotsize"> </td>
</tr>
<tr>
<td class="knotcross" rowspan=2>15</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/15a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/15a-hyp.csv.bz2">85,262 knots</a></td><td class="knotsizeupper">(1.6 MB)</td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/15n-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/15n-satellite.csv.bz2">6 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/15n-hyp.csv.bz2">168,023 knots</a></td><td class="knotsize">(4.0 MB)</td>
</tr>
<tr>
<td class="knotcross" rowspan=2>16</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/16a-hyp.csv.bz2">379,799 knots</a></td><td class="knotsizeupper">(7.9 MB)</td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/16n-torus.csv.bz2">1 knot</a></td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/16n-satellite.csv.bz2">10 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/16n-hyp.csv.bz2">1,008,895 knots</a></td><td class="knotsize">(27 MB)</td>
</tr>
<tr>
<td class="knotcross" rowspan=2>17</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/17a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/17a-hyp.csv.bz2">1,769,978 knots</a></td><td class="knotsizeupper">(41 MB)</td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/17n-satellite.csv.bz2">29 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/17n-hyp.csv.bz2">6,283,385 knots</a></td><td class="knotsize">(184 MB)</td>
</tr>
<tr>
<td class="knotcross" rowspan=2>18</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/18a-hyp.csv.bz2">8,400,285 knots</a></td><td class="knotsizeupper">(215 MB)</td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/18n-satellite.csv.bz2">86 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/18n-hyp.csv.bz2">39,866,095 knots</a></td><td class="knotsize">(1.3 GB)</td>
</tr>
<tr>
<td class="knotcross" rowspan=2>19</td><td class="knotalt">alternating</td>
<td class="knotcountuppersmall"><a href="http://www.maths.uq.edu.au/~bab/knots/19a-torus.csv.bz2">1 knot</a></td>
<td class="knotcountuppersmall">—</td>
<td class="knotcountupper"><a href="http://www.maths.uq.edu.au/~bab/knots/19a-hyp.csv.bz2">40,619,384 knots</a></td><td class="knotsizeupper">(1.1 GB)</td>
</tr><tr>
<td>non-alternating</td>
<td class="knotcountsmall">—</td>
<td class="knotcountsmall"><a href="http://www.maths.uq.edu.au/~bab/knots/19n-satellite.csv.bz2">245 knots</a></td>
<td class="knotcount"><a href="http://www.maths.uq.edu.au/~bab/knots/19n-hyp.csv.bz2">253,510,828 knots</a></td><td class="knotsize">(8.7 GB)</td>
</tr>
</table>
<p>
<i><a href="#contents">(Back to top...)</a></i>
<h2><a name="census">3-manifold census data</a></h2>
Regina ships with several different censuses of triangulations.
You can access most of these censuses by selecting
<i>File → Open Example</i> from Regina's main menu.
</ul>
<p>
Here you can download additional census files that are too
large to ship with Regina. You can also find the standard files
that <i>are</i> shipped, in case you have an older version of Regina
that did not include them.
<p>
You can open each of these data files directly within Regina.
Each file begins with a text packet that describes what
the census contains and where the data originally came from.
<p>
<table cellspacing=0 border=0 class="data"><tbody>
<tr>
<th class="first">Census</th>
<th>Origin</th>
<th>Download</th>
<th>Size (kB)</th>
</tr>
<tr><td class="subheading" colspan=4>Closed census</td></tr>
<tr>
<td class="first">All minimal triangulations of all closed orientable
prime 3-manifolds<br>
≤ 10 tetrahedra</td>
<td rowspan=3>Tabulated by
<a href="http://arxiv.org/abs/1101.3091">Burton</a></td>
<td><a href="census/closed-or-census.rga">closed-or-census.rga</a></td>
<td>848</td>
</tr>
<tr>
<td class="first">All minimal triangulations of all closed orientable
prime 3-manifolds<br>
≤ 11 tetrahedra <i>(too large to ship with Regina)</i></td>
<td><a href="census/closed-or-census-11.rga">closed-or-census-11.rga</a></td>
<td>1906</td>
</tr>
<tr>
<td class="first">All minimal triangulations of all closed non-orientable
P<sup>2</sup>-irreducible 3-manifolds<br>
≤ 11 tetrahedra</td>
<td><a href="census/closed-nor-census.rga">closed-nor-census.rga</a></td>
<td>537</td>
</tr>
<tr><td class="subheading" colspan=4>Closed hyperbolic census</td></tr>
<tr>
<td class="first">Smallest known closed hyperbolic 3-manifolds<br>
3000 orientable, 18 non-orientable</td>
<td rowspan=2>Tabulated
by <a href="http://projecteuclid.org/euclid.em/1048515809">Hodgson and
Weeks</a></td>
<td><a href="census/closed-hyp-census.rga">closed-hyp-census.rga</a></td>
<td>310</td>
</tr>
<tr>
<td class="first">Smallest known closed hyperbolic 3-manifolds<br>
11031 orientable, 18 non-orientable
<i>(too large to ship with Regina)</i></td>
<td><a href="census/closed-hyp-census-full.rga">closed-hyp-census-full.rga</a></td>
<td>1275</td>
</tr>
<tr><td class="subheading" colspan=4>Cusped hyperbolic census</td></tr>
<tr>
<td class="first">All minimal triangulations of all cusped hyperbolic
orientable 3-manifolds<br>
≤ 7 tetrahedra</td>
<td rowspan=4>Tabulated by
<a href="http://arxiv.org/abs/1405.2695">Burton</a></td>
<td><a href="census/cusped-hyp-or-census.rga">cusped-hyp-or-census.rga</a></td>
<td>354</td>
</tr>
<tr>
<td class="first">All minimal triangulations of all cusped hyperbolic
non-orientable 3-manifolds<br>
≤ 7 tetrahedra</td>
<td><a href="census/cusped-hyp-nor-census.rga">cusped-hyp-nor-census.rga</a></td>
<td>179</td>
</tr>
<tr>
<td class="first">All minimal triangulations of all cusped hyperbolic
orientable 3-manifolds<br>
≤ 9 tetrahedra <i>(too large to ship with Regina)</i></td>
<td><a href="census/cusped-hyp-or-census-9.rga">cusped-hyp-or-census-9.rga</a></td>
<td>7902</td>
</tr>
<tr>
<td class="first">All minimal triangulations of all cusped hyperbolic
non-orientable 3-manifolds<br>
≤ 9 tetrahedra <i>(too large to ship with Regina)</i></td>
<td><a href="census/cusped-hyp-nor-census-9.rga">cusped-hyp-nor-census-9.rga</a></td>
<td>3571</td>
</tr>
<tr><td class="subheading" colspan=4>Knot and link complements</td></tr>
<tr>
<td class="first">Christy's collection of knot complements
(≤ 11 crossings) and link complements (≤ 10 crossings)</td>
<td>Collected by Christy<br>Shipped with
<a href="http://www.ms.unimelb.edu.au/~snap">Snap 1.9</a></td>
<td><a href="census/hyp-knot-link-census.rga">hyp-knot-link-census.rga</a></td>
<td>132</td>
</tr>
</tbody></table>
<p>
In older versions of Regina, Christy's collection used to be called
“hyperbolic knot / link complements”;
however, it also contains some (but not all) non-hyperbolic cases.
It also contains the duplicate Perko pair.
<p>
<i><a href="#contents">(Back to top...)</a></i>
<h2><a name="weber-seifert">Weber-Seifert dodecahedral space</a></h2>
The <i>Weber-Seifert dodecahedral space</i> was one of the first-known
examples of a hyperbolic 3-manifold, and Thurston conjectured around 1980
that this space was non-Haken. A proof was obtained in 2009 using a
blend of theory and computation, and the details can be found in
the following paper:
<ul>
<li>B.B., J. Hyam Rubinstein and Stephan Tillmann,
<a href="http://arxiv.org/abs/0909.4625/"><i>The Weber-Seifert
dodecahedral space is non-Haken</i></a>,
Trans. Amer. Math. Soc. <b>364</b> (2012), no. 2, 911–932.</li>
</ul>
<p>
Because the proof involves computation, there is a fair amount of
supporting data, including the 23-tetrahedron triangulation of the
Weber-Seifert dodecahedral space and its 1751 standard vertex
normal surfaces. This is stored in a Regina data file, which you
can download here:
<ul>
<li><a href="files/weber-seifert.rga">Download weber-seifert.rga</a> (162 kb)
</ul>
<p>
<i><a href="#contents">(Back to top...)</a></i>
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