Knot data tables

Before using the tables understand carefully the notations and conventions!

After using the tables (in your paper) please put a reference to this URL. Thank you.

Please see my online talk recording for information on KnotScape.

Also please be aware that many tabulations here, despite requiring a laborious process, beg independent verification. Thus if you discover any inconsistencies, please report to me immediately. Similarly, please do if you made your own (or are aware of anyone else's) check, so that I can put a reference to it.


Knotation

A personal note. With the abundance of available tabulations, it becomes increasingly impossible to maintain an informative notation. I fixed my numbering long before the "a/n" nomenclature of KnotInfo (and a sequel of referring papers) became afloat. I took example specifically from 136714, that occurred in the Lickorish-Millett paper on the HOMFLY polynomial. After all, a single integer had been used as index for quite a while (at least since Alexander-Briggs), and this seemed to me the reasonable thing to continue doing...


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. (All of them have a (-)achiral mutant.)


Braid representatives and braid index

Here is a table of braid indices of prime non-alternating knots through 13 crossings.
(MWF is sharp for alternating knots through 17 crossings.) An astersik (*) indicates that only 2cMWF is sharp,
an ampersand sign (&) indicates that only 3cMWF is sharp.

Here are tables of braid representatives of prime knots through 13 crossings (updated Dec '21; "±i" stands for σi±1 with the Artin generators σi). Here is a list of the prime knots of 14-16 crossings of braid index 2,3. Here are non-alternating ones of braid index 4.
(The last two previously undecided knots, 161059154 and 161153788, have been ruled out using 4-cable HOMFLY.)
Braid index 4 is determined by MWF for alternating knots (see here).

Also, the braid index of all strongly quasi-positive prime knots up to 16 crossings has been determined (see section below).


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.


Positive knots

Here is the list of non-alternating (prime) positive knots up to 13 (232 knots) and of 14-15 crossings (3355 knots); positive diagram in {}.

Alternating ones are easy to test (for example σ= 2max deg Δ or min deglP=max degmP), so are not given.


Strongly quasi-positive knots

Here is the list of non-alternating (prime) strongly quasi-positive knots up to 16 crossings (22,009 knots). Here are band representations, where [ij] (for j-i>1) is the positive band between and above (upward-oriented) strands i,j, so "i (i+1) ... (j-2) (j-1) (2-j)...-i" in Artin generators (e.g. [13]="1 2 -1" and "2"=[23]).

Also, for all knots we now know that a minimal string positive band representation exists (answering a question of Rudolph), and is featured in the above list. The last knot standing, until May 2023, was 161057125, of genus 4, where we have a 6-braid positive band representation, but braid index 5 could not be ruled out. This knot was extremely hard to resolve, and now merits its own page.

Alternating ones are easy to test (same condition as for positive), so we did not write them. The method of obtaining (minimal string) positive band representations for them is explained in my paper "Realizing strongly quasipositive links and Bennequin surfaces" on my papers list.


Quasi-positive knots

The quasi-positive knots are far harder to decide, because essentially one has to determine the (smooth) 4-genus rather than the genus. The following is the list up to 13 crossings.

I resolved a question of here.

Theorem. Every quasi-positive alternating knot is special alternating.

Thus quasi-positive knots can be split into
(a) special alternating,
(b) strongy quasi-positive non-alternating
(c) quasi-positive (not strongy quasi-positive) slice
(d) quasi-positive not strongy quasi-positive not slice
(Distinguishing between slice and not is merely a technical detail of how I managed the lists, but once I separated them, I thought it's better not to mix them again.)

For completeness I add family (a), but you can get (b) from the lists above. For (c) and (d), the already given braid representative is quasi-positive (though it is sometimes a bit tricky to see that).

up to 12 crossings 13 crossings
special alternating 156 knots 309 knots
sqp non-alternating 108 knots 168 knots
qp (not sqp) slice 16 knots 26 knots
qp not sqp not slice 183 knots 726 knots*

* = plus possibly some of the following 3 yet undecided: 136724, 137374, 138494.

Some information on 4-genus can be found on KnotInfo.


This page was last updated Thu Feb 1 07:43:23 KST 2024.

Alexander Stoimenow,
stoimeno_stoimenov.net