Knot data tables

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

• The notation for knots up to 10 crossings is this of Rolfsen's book. However, invariants were computed using KnotScape. KnotScape uses a different numbering! Here is how to convert KnotScape to Rolfsen and Rolfsen to KnotScape numbering.

  While the importance (and popularity) of the book has established this numbering as the inevitable standard, it turns out that nothing therein is the original. As can be seen from Conway's paper (reference below), Rolfsen has added numbers to the 10 crossing knots in the order they appeared there (incl. Perko's dup). This overrides Tait's numbering for the alternating knots and abolishes Little's nomenclature for the non-alternating ones. Alexander-Briggs (which Rolfsen conforms to) had earlier done the same for 9 or fewer crossings.

• Rolfsen's last four 10 crossing knots have been renumbered, to avoid the Perko duplication. A further mistake in Rolfsen's tables is that therein 10₈₃ and 10₈₆ were swopped: the Conway notation and Alexander polynomial for each one referred to the diagram of the other. We exchange not diagrams, but Alexander polynomial and Conway notation to fix the mistake (like the table in the new 2003 edition of the Rolfsen book, Dror Bar-Natan's Knot Atlas, and Chuck Livingston's Table of Knot Invariants, and unlike in Kawauchi's book!).

• On this page, throughout, in KnotScape's numbering, non-alternating knots are appended after alternating ones of the same crossing number. Thus, for example, Rolfsen's 10₁₆₄ (a.k.a., 10₁₆₅, before Perko's duplication is removed) is KnotScape's 10₁₆₁, which means (at the presence of 123 alternating 10 crossing knots) KnotScape's 38^th non-alternating 10 crossing knot.

• The notation for knots of 11-16 crossings is this of KnotScape; always non-alternating knots are appended after alternating ones of the same crossing number. Thus, for example, 13₄₈₇₈=13a4878 is the last alternating 13 crossing knot (=(2,13) torus knot), 13₄₈₇₉=13n1 is the first non-alternating.

• The original (pre-KnotScape) tables of 11 crossing knots are available on Perko's page, with references to Conway and Caudron available on Sloane's OEIS.

  For reference, here's my proposed (scan-check-) "cleaned up" list of Perko's 11 crossing paper (with notes what I fixed).

• A dictionary KnotScape to Perko and Perko to KnotScape was extracted from KnotInfo's "Conway name". However, KnotScape's 11₇₁, 11₇₂ had Conway names (303,305) mixed up (cf. the error below); this is fixed here.

  Early on addicted to the electronic KnotScape list, I always have a bad conscience especially about 11 crossings, since a lot of work (in particular of Caudron) eludes proper acknowledgement.

• Attributing the 11 crossing "Conway name" numbering (solely) to Conway is somewhat misleading.

  The numbering for alternating knots was mainly due to Little. Conway reproduced Little's numbers but reordered his list. The 11 omissions (and 1 duplication) Conway found are not used any longer the way Conway numbered them, but the way Perko later did. (Conway's 11₄₀₀, 11₄₀₁, 11₄₀₂, ..., 11₄₁₁ are Perko's 11₁₄₁, 11₁₄₂, 11₃₅₈, ..., 11₃₆₇.)

  The numbering for non-alternating knots was introduced by Perko. While it may follow Conway's list, there at least are 4 omissions (2 found by Lombardero, 2 by Caudron) fixed in Perko's paper.

  Perko's paper lists further proposed corrections to Conway (including his - confirmed - mixup of 11₃₀₃, 11₃₀₅).

• Smith, Schermann, and Rankin (see here) had a computer tool to generate prime alternating knot tables that I used at the time (i.e., around 2005) to get the list of 17 and 18 crossings.

• A further table, up to 19 crossings (knots wherein are not used on this page), was published recently by Burton here.

• Some information on knot tables is also available on Wolfram.
  

Knot invariants

• The paper form of the tables contains further references, details on the conventions (with skein relations) and notation (with examples how to read it). Please check it out, if the below points leave you unsure you understand the tables properly.

• The knot is mirrored as KnotScape draws it. Thus some invariants refer to the obverse pictures of those of Rolfsen. Here is how to convert mirroring between both tables, where the knot is written as in Rolfsen's table. For example, Rolfsen's 10₅ is KnotScape's 10₅₆, and in both tables the knot is differently mirrored. (Thus, better speaking, Rolfsen's 10₅ is KnotScape's !10₅₆.)

  An asterisk means that the knot is amphicheiral. Particular care is needed with (Rolfsen's) 10₇₁, for it is not amphicheiral, yet no invariant tabulated here (and computable with KnotScape) can distinguish it from its mirror image.

• The notation of polynomials is a combination of those of Lickorish-Millett and Adams. For the Jones and Alexander polynomial, if the absolute term occurs between its minimal and maximal degrees, then it is bracketed, else the minimal degree is recorded in braces before the coefficient list. For example, the Jones polynomial of 8₁₉ is t³+t⁵-t⁸. For the Conway and Brandt-Lickorish-Millett-Ho polynomials the minimal degree is not indicated, as it is always zero. For the Conway polynomial, odd degree (zero) coefficients are omitted.

• For the HOMFLY polynomial we omit entries of l^km^s with one of k, s odd (as they are zero). For the Kauffman polynomial monomials a^kz^s only for even parity of k + s occur. The other coefficients are omitted or for k = 0 replaced by an asterisk to make clear the degrees of a. Our convention for the HOMFLY and Kauffman polynomials differs from this of Lickorish-Millett resp. Kauffman by interchanging l with l^-1 and a with a^-1.

• Then the signature is tabulated in the convention that the positive (right-hand) trefoil has signature 2. (Positive links have positive signature!)

• Finally, we give the values of the degree-3-Vassiliev invariant v₃ in its asymmetric (changing the signunder mirroring) additive (under connected sum) version. The normalization is so that v₃=1 on the positive trefoil. This is is the convention of Polyak-Viro. (Fiedler's normalization differs by a factor of 4.)

Mutations

Up to 13 crossings

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

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

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

• Braid representatives are of minimal (=braid index) width.

• They are chosen of (shortest) minimal braid length among minimal (width) braid representatives. (Note that some knots may not have minimal width braid representatives of minimal length or vice versa, for example 10₁₃₆!)

• The notation for knots through 10 crossings is this of Rolfsen's book, for 11 and 12 crossings this of KnotScape; the Perko duplication and the mistake of 10₈₃ and 10₈₆ are handled as said above.

• Care is not taken how the knot is mirrored!

• The braid representatives of 12₁₃₁₆, 12₁₃₁₉ and 12₁₄₁₇ were corrected following a comment of Chuck Livingston.

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

Genera / Canonical genera

• For up to 10 crossings and for alternating knots genus and canonical genus are the same, and given by the degree of the Alexander polynomial.

• For all knots up to 13 crossings canonical genus is given by half the degree of the Alexander variable in the skein polynomial (Morton's inequality is sharp).

• If the degree of the Alexander polynomial coincides with half the degree of the Alexander variable in the skein polynomial, then genus is equal to canonical genus; there are 49 non-alternating 11 and 12 crossing knots and 287 non-alternating 13 crossing knots where this is not the case.

• Of the 49 knots of 11 and 12 crossings, 33 have still minimal genus canonical surfaces, proved by Gabai's disc decomposition. Similarly do 176 of the 287 non-alternating 13 crossing knots. These knots are marked with an asterisk.

• The remaining 16 knots of 11 and 12 crossings and 111 knots of 13 crossings are marked with an exclamation mark.

• The remaining 11 and 12 crossing knots all have genus 2 and canonical genus 3. Minimal genus surfaces can be found by Rudolph's braided surface construction. The minimality follows from the Alexander polynomial (it has degree 2), except for 3 knots.

• The 3 knots left are 11₄₀₉ (the K-T knot), 12₁₃₁₁ and 12₁₄₁₂. The first knot is known to have a genus two disc decomposable surface by Gabai. Genus 2 braided surfaces for the others were found (stably) Hopf plumbing equivalent, and a genus 2 braided surface for 12₁₄₁₂ was found disc-annulus decomposable by Mikami Hirasawa, so these knots are also of genus 2. Later I observed that they also follow (much more easily) from a result of Kobayashi, since they have unknotting number 1.

• The 111 knots of 13 crossings are subjected to the same procedure. First we find braided surfaces which are candidates for minimal genus. Then we use the Alexander polynomial, Kobayashi's result (which says that genus ≥2 when unknotting number =1), and Hopf plumbing equivalence test. Latter means that we tried to relate the braided surfaces by Hopf (de)plumbing to minimal genus surfaces of other knots we know. This settled 106 of the knots.

• The 5 remaining 13 crossing knots are 13₅₇₆₄, 13₅₈₂₆, 13₅₈₄₁, 13₅₈₉₇ and 13₈₇₆₅. For them I could only expect that the upper genus bound is exact, and I could establish that the projected genus=3 knots (13₅₇₆₄,13₈₇₆₅) are Hopf plumbing equivalent (so genus=3 for one would settle the other).

• These 5 knots were later settled by a calculation of Stefan Friedl, Nathan Dunfield, Nicholas Jackson, and Stefano Vidussi using twisted Alexander polynomials. This is the only proof I know so far for them. (Friedl and Taehee Kim had confirmed also the 11 and 12 crossing knots for which the Alexander polynomial does not give the genus.)

• Our table for the 11 crossing knots confirms a computation done previously by Jake Rasmussen, using the Ozsvath-Szabo knot Floer homology.

Fibering

• It is known that 10 or less crossing knots are fibered if (and only if) the Alexander polynomial is monic.

• It was checked with Mikami Hirasawa that 11 crossing knots are fibered if (and only if) the Alexander polynomial is monic and of degree matching the genus. (Cf. the K-T and Conway knot have the trivial polynomial, and 11₄₄₀ has monic polynomial of smaller degree.)

• The fiberedness of 12 crossing knots is determined as follows. Thirteen 12 crossing knots with monic Alexander polynomial of degree matching the genus are non-fibered. Friedl and Kim calculated this by twisted Alexander polynomials (see here), and later Jake Rasmussen reportedly made independent verification using the Ozsvath-Szabo knot Floer homology. (If you have a different verification, please report to me. Using Gabai's methods, so far I checked myself the 8 knots where minimal genus surfaces are canonical.) These knots are marked with a dollar sign. The other knots with monic Alexander polynomial of degree matching the genus are fibered.

• Now also the fiberedness problem for 13 crossing knots is settled. There are 2759 knots with monic Alexander polynomial of degree = genus AND genus>1. (Fibered genus 1 knots are only the trefoil and figure-8-knot.) Of them 78 knots are non-fibered. These knots are marked with a dollar sign. The other 2681 knots with monic Alexander polynomial of degree matching the genus are fibered. The decision (except for 15 knots) uses Gabai's disk decomposition of canonical surfaces (when canonical surfaces of minimal genus exist) and the plumbing equivalence test to simpler knots (when canonical surfaces of minimal genus do not exist). This gives a result except for 15 knots of the latter type. I could not settle these myself and consulted S. Friedl, who reported them non-fibered, following his calculation with Dunfield/Jackson/Vidussi. Apparently the other knots are not yet fully independently verified by them.

Genus generators

Amphicheiral knots

Positive knots

Alternating knots are easy to test for positivity (for example σ= 2max deg Δ or min deg_lP=max deg_mP), so they are not given here.

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 16_1057125, 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.

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 2 yet undecided: 13₇₃₇₄, 13₈₄₉₄.

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

• λ-numbers are the negated maximal Thurston-Bennequin invariants. (Rudolph)

• For alternating knots, λ-numbers are given by Kauffman polynomial degrees 1-min deg_aF (for the knot), 1+max deg_aF (for the mirror image). (It was proved by Toshifumi Tanaka.) In general these degrees give lower bounds.

• For knots of 13 or fewer crossings, the maximal Thurston-Bennequin invariants (i.e., with the opposite sign to the numbers given here) are recorded on the KnotInfo site.

• The sum of the two λ-numbers of mirror images is the arc index. (It was proved by Dynnikov-Prasolov.)

• You can identify mirror images thus. When the invariants for both mirror images are not equal, the Kauffman polynomial lower bounds (see above) identify the mirror image except for 161 knots. For all these 161 knots, the λ-numbers of the mirror images are 6 and 7, resp. (and all knots have arc index 13). To sort mirror images out, I append to the list of λ-numbers minimal grids for these 161 knots. ( Alternatively you can retrieve the grid diagrams also from here.) From the minimal grids one can determine the λ-numbers (as proved by Dynnikov-Prasolov). All these 161 knots have (interestingly) signature σ=±2, and (λ(K),λ(!K))=(6,7) exactly when σ(K)=+2, so mirror images can also be distinguished this way.