Regina - Supporting Data

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

Regina now includes native support for knots and links. If you wish to play with some smaller examples, you can open Regina and select Open Example → Prime Knots from the menu.

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.

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:

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
Christy's collection of knot complements (≤ 11 crossings) and link complements (≤ 10 crossings) Collected by Christy
Shipped with Snap 1.9
christy-knots-links.rga 132

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.

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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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