Regina - Supporting Data

Knot tables
3-manifold census data
Weber-Seifert dodecahedral space
Related articles
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Knot tables

The next release of Regina (version 5.2) will include native support for knots and links. To accompany this, you can (right now) download the tables of all 352,152,252 prime non-trivial knots with up to 19 crossings.

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 bzip2. The fields include:

Citation: If you wish to cite this data, please reference Benjamin A. Burton, “The next 350 million knots”, which should appear on the arXiv in June 2018.

Download all 3–12 crossing knots at once (56 kB)
Download all 13–16 crossing knots at once (41 MB)
Download individual tables (up to 19 crossings) below:

Crossings Torus Satellite Hyperbolic
3alternating 1 knot  
4alternating 1 knot 
5alternating 1 knot 1 knot 
6alternating 3 knots 
7alternating 1 knot 6 knots 
8alternating 18 knots 
non-alternating 1 knot 2 knots 
9alternating 1 knot 40 knots 
non-alternating 8 knots 
10alternating 123 knots 
non-alternating 1 knot 41 knots 
11alternating 1 knot 366 knots 
non-alternating 185 knots 
12alternating 1,288 knots 
non-alternating 888 knots 
13alternating 1 knot 4,877 knots 
non-alternating 2 knots 5,108 knots 
14alternating 19,536 knots 
non-alternating 1 knot 2 knots 27,433 knots 
15alternating 1 knot 85,262 knots(1.6 MB)
non-alternating 1 knot 6 knots 168,023 knots(4.0 MB)
16alternating 379,799 knots(7.9 MB)
non-alternating 1 knot 10 knots 1,008,895 knots(27 MB)
17alternating 1 knot 1,769,978 knots(41 MB)
non-alternating 29 knots 6,283,385 knots(184 MB)
18alternating 8,400,285 knots(215 MB)
non-alternating 86 knots 39,866,095 knots(1.3 GB)
19alternating 1 knot 40,619,384 knots(1.1 GB)
non-alternating 245 knots 253,510,828 knots(8.7 GB)

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3-manifold census data

Regina ships with several different censuses of triangulations. You can access most of these censuses by selecting File → Open Example from Regina's main menu.

Here you can download additional census files that are too large to ship with Regina. You can also find the standard files that are shipped, in case you have an older version of Regina that did not include them.

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.

Census Origin Download Size (kB)
Closed census
All minimal triangulations of all closed orientable prime 3-manifolds
≤ 10 tetrahedra
Tabulated by Burton closed-or-census.rga 699
All minimal triangulations of all closed orientable prime 3-manifolds
≤ 11 tetrahedra (too large to ship with Regina)
closed-or-census-11.rga 1906
All minimal triangulations of all closed non-orientable P2-irreducible 3-manifolds
≤ 11 tetrahedra
closed-nor-census.rga 389
Closed hyperbolic census
Smallest known closed hyperbolic 3-manifolds
3000 orientable, 18 non-orientable
Tabulated by Hodgson and Weeks closed-hyp-census.rga 310
Smallest known closed hyperbolic 3-manifolds
11031 orientable, 18 non-orientable (too large to ship with Regina)
closed-hyp-census-full.rga 1275
Cusped hyperbolic census
All minimal triangulations of all cusped hyperbolic orientable 3-manifolds
≤ 7 tetrahedra
Tabulated by Burton cusped-hyp-or-census.rga 354
All minimal triangulations of all cusped hyperbolic non-orientable 3-manifolds
≤ 7 tetrahedra
cusped-hyp-nor-census.rga 179
All minimal triangulations of all cusped hyperbolic orientable 3-manifolds
≤ 9 tetrahedra (too large to ship with Regina)
cusped-hyp-or-census-9.rga 7902
All minimal triangulations of all cusped hyperbolic non-orientable 3-manifolds
≤ 9 tetrahedra (too large to ship with Regina)
cusped-hyp-nor-census-9.rga 3571
Knot and link complements
All hyperbolic knot complements (≤ 11 crossings) and link complements (≤ 10 crossings) Tabulated by Christy
Shipped with Snap 1.9
hyp-knot-link-census.rga 132

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Weber-Seifert dodecahedral space

The Weber-Seifert dodecahedral space 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:

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:

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Related Articles

The following papers describe some of the algorithms that Regina implements.

Burton's PhD thesis contains more detailed descriptions of some of the topological structures, concepts and algorithms used in Regina. You can download it from his website.

This list is by no means complete. For more relevant papers, see the bibliography in the handbook, or Regina's summary article in Experimental Mathematics.

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