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

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 nonalternating ones.
AlexanderBriggs (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
10_{83} and 10_{86} 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
BarNatan'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, nonalternating
knots are appended after alternating ones of the same
crossing number. Thus, for example, Rolfsen's 10_{164}
(a.k.a., 10_{165}, before Perko's
duplication is removed) is KnotScape's 10_{161},
which means (at the presence of 123 alternating 10 crossing knots)
KnotScape's 38^{th} nonalternating 10 crossing knot.

The notation for knots of 1116 crossings is this of KnotScape; always
nonalternating knots are appended after alternating ones of the same crossing number.
Thus, for example, 13_{4878}=13a4878 is the last alternating 13
crossing knot (=(2,13) torus knot),
13_{4879}=13n1 is the first nonalternating.
 The original (preKnotScape) 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 (scancheck)
"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
11_{71}, 11_{72} 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 11_{400}, 11_{401}, 11_{402}, ...,
11_{411} are Perko's 11_{141}, 11_{142},
11_{358}, ..., 11_{367}.)
The numbering for nonalternating 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 11_{303}, 11_{305}).

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.
A personal note. With the abundance of available tabulations,
it becomes increasingly impossible to specify knots informatively
and omnicongenially. I fixed my numbering long before the "a/n" nomenclature
was popularized by resources like KnotInfo, and a string of referring papers. I took example
specifically from 13_{6714}, that occurred in the LickorishMillett (1987!)
paper
on the HOMFLY polynomial (Example 16 and Fig 37).
After all, a single integer had been used as index for quite a while
(at least since AlexanderBriggs), and this seemed to me the
reasonable thing to continue doing... This means that my numbering
is not an artifact of my imagination, and am not "stubborn" and
"userunfriendly" by sticking to a convention that others choose not to
share.
Knot invariants
Here are tables of
Alexander,
Conway,
Jones,
Homfly,
BrandtLickorishMillettHo and
Kauffman polynomials,
the degree3Vassiliev invariant and
signatures of knots with up to
10 crossings.

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 10_{5}
is KnotScape's 10_{56}, and in both tables the
knot is differently mirrored. (Thus, better speaking,
Rolfsen's 10_{5} is KnotScape's !10_{56}.)
An asterisk means that the knot is amphicheiral.
Particular care is needed with (Rolfsen's) 10_{71},
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
LickorishMillett 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 8_{19} is t^{3}+t^{5}t^{8}.
For the Conway and BrandtLickorishMillettHo 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^{k}m^{s} with one of k, s odd
(as they are zero). For the Kauffman polynomial monomials
a^{k}z^{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 LickorishMillett resp. Kauffman by interchanging l
with l^{1} and a with a^{1}.

Then the signature is tabulated in the convention that the positive
(righthand) trefoil has signature 2. (Positive links have
positive signature!)

Finally, we give the values of the degree3Vassiliev invariant
v_{3} in its asymmetric (changing the signunder mirroring)
additive (under connected sum) version. The normalization is so that
v_{3}=1 on the positive trefoil. This is is the convention of
PolyakViro. (Fiedler's normalization differs by a factor of 4.)
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:
(14_{41721},14_{42125})
(14_{41739},14_{42126})
(14_{41763},14_{42021})
(14_{42947},14_{43476})
(14_{42953},14_{43572})
(14_{43904},14_{46158})
(14_{43907},14_{46187})
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 lowindexsubgroups of the
fundamental group of the 2fold 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 nonalternating 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 σ_{i}^{±1}
with the Artin generators σ_{i}).
 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 10_{136}!)

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 10_{83} and 10_{86} are handled
as said above.

Care is not taken how the knot is mirrored!
 The braid representatives of 12_{1316},
12_{1319} and 12_{1417} were corrected
following a comment of Chuck Livingston.
Here is a list of the prime
knots of 1416 crossings of braid index
2,3. Here are nonalternating ones
of braid index 4.
(The last two previously undecided knots,
16_{1059154} and 16_{1153788}, have been ruled
out using 4cable HOMFLY.)
Braid index 4 is determined by MWF for alternating knots
(see here).
Also, the braid index of all strongly quasipositive
prime knots up to 16 crossings has been determined (see section below).
Genera / Canonical genera
This table shows genera (and canonical genera)
of nonalternating prime 11 and 12 crossing knots, and of
the 13 crossing nonalternating knots.
 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 nonalternating 11 and
12 crossing knots and 287 nonalternating 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 nonalternating 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 11_{409} (the KT knot), 12_{1311}
and 12_{1412}. 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 12_{1412} was found discannulus
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 13_{5764},
13_{5826}, 13_{5841}, 13_{5897} and
13_{8765}. For them I could only expect that the upper
genus bound is exact, and I could establish that the projected
genus=3 knots (13_{5764},13_{8765}) 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 OzsvathSzabo 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 KT and Conway knot have the trivial polynomial, and
11_{440} 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 nonfibered. Friedl and Kim calculated this by
twisted Alexander polynomials (see
here), and later
Jake Rasmussen reportedly made independent verification using the
OzsvathSzabo 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
figure8knot.) Of them 78 knots are nonfibered. 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 nonfibered, following his
calculation with Dunfield/Jackson/Vidussi. Apparently the other
knots are not yet fully independently verified by them.
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 nonspecial (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 nonalternating (prime)
positive knots up to 13 (232 knots)
and of 1415 crossings (3355 knots);
positive diagram in {}.
(These positive diagram knot identifiers have no intended
meaning; again, they just came about by the technical algorithm
used to construct the list. I consistently manage diagrams in the form
<crossing number> <id> <DT sequence>, so an <id> always goes along...)
Alternating knots are easy to test for positivity (for example σ=
2max deg Δ or min deg_{l}P=max deg_{m}P),
so they are not given here.
Strongly quasipositive knots
Here is the list of nonalternating (prime) strongly quasipositive
knots up to 16 crossings (22,009 knots).
Here are band representations,
where [ij] (for ji>1) is the positive band between and
above (upwardoriented) strands i,j, so
"i (i+1) ... (j2) (j1) (2j)...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 16_{1057125}, of genus 4, where we
have a 6braid 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.
Quasipositive knots
The quasipositive knots are far harder to decide, because essentially
one has to determine the (smooth) 4genus rather than the genus.
The following is the list up to 13 crossings.
I resolved a question of here.
Theorem. Every quasipositive alternating knot is special alternating.
Thus quasipositive knots can be split into
(a)
special alternating,
(b) strongy quasipositive nonalternating
(c) quasipositive (not strongy quasipositive) slice
(d) quasipositive not strongy quasipositive 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 quasipositive (though it is sometimes a bit tricky to see that).
* = plus possibly some of the following 2 yet undecided:
13_{7374}, 13_{8494}.
Some information on 4genus can be found on
KnotInfo.
Arc index and maximal ThurstonBennequin invariant
The following is a list of the
λnumbers of prime nonalternating 14 crossing knots,
with mirror images.
This is the minimal framing of an annulus around the knot
which is strongly quasipositive.

λnumbers are the
negated maximal ThurstonBennequin invariants.
(Rudolph)

For alternating knots, λnumbers are
given by Kauffman polynomial degrees 1min deg_{a}F (for the knot),
1+max deg_{a}F (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 ThurstonBennequin invariants
(i.e., with the opposite sign to the numbers given here) are
recorded on the
KnotInfo site.

The sum of the two λnumbers of mirror images
is the arc index. (It was proved by DynnikovPrasolov.)

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 λnumbers 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 λnumbers minimal grids
for these 161 knots.
(
Alternatively you can retrieve the grid diagrams also from
here.)
From the minimal grids one
can determine the λnumbers (as proved by DynnikovPrasolov).
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.
This page was last updated Sun Jul 28 21:55:37 KST 2024.
Alexander Stoimenow,
stoimeno_stoimenov.net