Knot data tables

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Knotation


Knot invariants

Here are tables of Alexander, Conway, Jones, Homfly, Brandt-Lickorish-Millett-Ho and Kauffman polynomials, the degree-3-Vassiliev invariant and signatures of knots with up to 10 crossings.


Mutations

Up to 13 crossings

Here is a list of all mutant groups (91 groups, consisting of 86 pairs and 5 triples) of prime knots through 12, and of 13 crossings (774 groups, consisting of 703 pairs, 38 triples, 32 groups of 4 and one group of 6).

To find the mutants, first I determined the groups of knots with equal volume, Alexander polynomial and Jones polynomial. Then mutations were explicitly exhibited in minimal crossing diagrams.

Care is not taken how the knots are mirrored!

The list of groups through 12 crossings confirms, at least in length, a compilation done shortly previously by hand by David De Wit, see here.

14 crossings

Here is the list of 14 crossing mutant groups (without DT codes): 4435 total groups, consisting of 3917 pairs, 233 triples, 262 quadruples, 17 groups of 6, and 6 groups of 8 knots.

The list was determined with the above type of check. It left over 7 pairs:

(1441721,1442125)
(1441739,1442126)
(1441763,1442021)
(1442947,1443476)
(1442953,1443572)
(1443904,1446158)
(1443907,1446187)

The knots in the first and third pair are distinguished by the Whitehead double skein polynomial. The second pair was the most difficult to break. Daniel Matei distinguished it by abelianizations of kernels of homomorphisms or low-index-subgroups of the fundamental group of the 2-fold branched cover. The last 4 pairs are mutants, but display the mutation only in 15 crossing diagrams.

15 crossings

Here is the list of 15 crossing mutant groups (without DT codes): 29049 total groups, consisting of 24884 pairs, 1000 triples, 2909 quadruples, 172 groups of 6, and 84 groups of 8 knots.

There are 78 pairs (marked with an asterisk) displaying the mutation only in 16 crossing diagrams, and 6 pairs only in 17 crossing diagrams (marked with an exclamation sign). There are 34 more pairs with equal volume, Alexander polynomial and Jones polynomial. 14 of them are distinguished by the Whitehead double skein polynomial, and the other 20 by the aforementioned group theoretic tests.

Mirror images

Since I ignored distinction between knots and mirror images, I have not listed chiral knots which are mutants to their mirror images. Such knots do not occur up to 13, and for 15 crossings, but there are 13 knots of 14 crossings. They are here.


Braid descriptions

Here are tables of braid descriptions of prime knots through 12 crossings. Here are braid descriptions of prime 13 crossing knots, not minimized (neither on width nor length).

Genera / Canonical genera

This table shows genera (and canonical genera) of non-alternating prime 11 and 12 crossing knots, and of the 13 crossing non-alternating knots.

Fibering


Genus generators

Here is the list of generators of genus 2 (24 knots), genus 3 (4017 knots), and (in gzipped format) genus 4, divided between special (1,480,238 knots, 22MB) and non-special (1,934,581 knots, 25MB), ones. The meaning of "generators" is described in my paper "Knots of (canonical) genus two" on my papers list. Briefly speaking, it means that the generators "generate" all prime knot diagrams of that (canonical) genus by flypes, crossing changes and a version of a full twist. (Note: The knot identifiers have no special meaning, and no relation to existing knot tables; they just came about by the technical procedures used to generate the lists.)


Amphicheiral knots

Here is the list of amphicheiral prime knots up to 16 crossings.


Alexander Stoimenow,
stoimeno_mathsci.kaist.ac.kr