Knot data tables
Before using the tables understand carefully the notations
After using the tables (in your paper) please put a
reference to this URL. Thank you.
The notation for knots is this of
Rolfsen's book. However, invariants were computed using
KnotScape uses a different numbering!
Here is how to convert
KnotScape to Rolfsen and
Rolfsen to KnotScape numbering.
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
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,
Bar-Natan's Knot Atlas, and
Livingston's Table of Knot Invariants, and unlike in
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 38th non-alternating 10 crossing knot.
For more on tables, see also
Here are tables of
the degree-3-Vassiliev invariant and
signatures of knots with up to
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 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
lkms with one of k, s odd
(as they are zero). For the Kauffman polynomial monomials
akzs only for even parity of k + s occur. The
other coefficients areomitted 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
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.)
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 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:
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.
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.
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 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).
- Braid descriptions are of minimal (=braid index) width,
if up to 18 crossings, of minimal braid length among minimal
(width) braid descriptions. (Note knots may not
have minimal width braid descriptions 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 descriptions of 121316,
121319 and 121417 were corrected
following a comment of Chuck Livingston.
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.
- For up to 10 crossings and for alternating knots genus and
canonical genus are the same, and given by the degree of the
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
- 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.
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
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.)
Here is the list of amphicheiral prime knots
up to 16 crossings.