Braids and Trees

What follows is a complete list of all prime knots of of genus 6 or less that are closures of positive braids, or arise as arborescent knots from a plane weighted tree all of whose weights are +2.

The list has been compiled for the paper [1].

If you have any questions concerning the list, feel free to email me. If you use the list for a paper, I would be grateful if you referenced this website, or the paper [1].

For references and more details, see below the table.

1. Braid
2. Number
3. Brick diagram / Checkerboard graph 4. g
5. sig
6. det
7. Alexander pol. 8. Sym.
9. Inv.
10. Mutants
11. Torus or Satellite
a3
31
1
2
3
1 -1D1
y
T(2,3)
a5
51
2
4
5
1 -1 1D1
y
T(2,5)
a7
71
3
6
7
1 -1 1 -1D1
y
T(2,7)
a3ba3b
819
3
6
3
1 -1 0 1D1
y
T(3,4)
a9
91
4
8
9
1 -1 1 -1 1D1
y
T(2,9)
a5ba3b
10124
4
8
1
1 -1 0 1 -1D1
y
T(3,5)
a4ba3b2
10139
4
6
3
1 -1 0 2 -3D2
y
a3b2a2b3
10152
4
6
11
1 -1 -1 4 -5D1
y
a2b2acb3c2
11n77
4
6
27
1 -1 -2 8 -11D2
y
a11
11a367
5
10
11
1 -1 1 -1 1 -1D1
y
T(2,11)
a7ba3b
12n242
5
8
1
1 -1 0 1 -1 1D1
y
a5ba4b2
12n472
5
8
13
1 -1 0 2 -4 5D1
y
a6ba3b2
12n574
5
8
9
1 -1 0 2 -3 3D2
y
a5b2a2b3
12n679
5
8
25
1 -1 -1 4 -6 71
n
a4b2a3b3
12n688
5
8
37
1 -1 -1 5 -9 111
n
a5ba5b
12n725
5
8
5
1 -1 0 1 -2 3D2
y
a3b3a3b3
12n888
5
8
45
1 -1 -1 6 -11 13D4
y
a3b2a2bcb3c
13n241
5
8
45
1 -1 -2 7 -10 11D1
y
A
a3b2a2bcb2c2
13n300
5
8
45
1 -1 -2 7 -10 11Z2
n
A
a3ba2b2cb3c
13n604
5
8
9
1 -1 -1 3 -2 1D1
y
a3b3a2cb3c
13n981
5
8
93
1 -1 -3 12 -21 25D1
y
B
a3b2a2cb3c2
13n1104
5
8
93
1 -1 -3 12 -21 251
n
B
a3ba3bcb3c
13n1176
5
8
21
1 -1 -1 4 -5 5D1
y
a4b3acb3c
13n1291
5
8
57
1 -1 -2 8 -13 15D2
y
C
a4b2acb3c2
13n1320
5
8
57
1 -1 -2 8 -13 15D1
y
C
a3b3acb3c2
13n2405
5
8
81
1 -1 -2 10 -19 23D2
y
a2b2acb2a2bcb
13n4587
5
6
7
1 -1 0 0 1 -1D7
y
(2,7) cable of the trefoil
ab2acb2acb3c
13n5016
5
6
15
1 -1 0 0 3 -5D2
y
a13
13a4878
6
12
13
1 -1 1 -1 1 -1 1D1
y
T(2,13)
14n56446
8
189
1 -1 -5 22 -43 53D2
y
a9ba3b
14n6022
6
10
3
1 -1 0 1 -1 1 -1D1
y
a7ba4b2
14n12201
6
10
23
1 -1 0 2 -4 5 -5D1
y
a6ba5b2
14n15856
6
10
27
1 -1 0 2 -4 6 -7D1
y
a8ba3b2
14n18079
6
10
15
1 -1 0 2 -3 3 -3D2
y
a7b2a2b3
14n20185
6
10
39
1 -1 -1 4 -6 7 -71
n
a5b2a4b3
14n20441
6
10
75
1 -1 -1 5 -10 15 -171
n
a6b2a3b3
14n20613
6
10
63
1 -1 -1 5 -9 12 -131
n
a7ba5b
14n21324
6
10
11
1 -1 0 1 -2 3 -3D1
y
a5ba3b2a2b
14n21881
6
8
1
1 -1 0 1 -1 0 1D1
y
T(3,7)
a5ba5b3
14n24551
6
10
35
1 -1 0 2 -5 8 -9D2
y
a4ba3b2a2b2
14n24763
6
8
9
1 -1 0 1 0 -2 3D1
y
a5b2a2b5
14n27110
6
10
51
1 -1 -1 4 -7 10 -11D1
y
a5b3a2b4
14n27128
6
10
71
1 -1 -1 5 -10 14 -151
n
a3b2a2b2a2b3
14n27179
6
8
21
1 -1 0 0 2 -4 5D1
y
a3b2a2b3a2b2
14n27214
6
8
25
1 -1 0 0 2 -5 7D2
y
a4b3a3b4
14n27233
6
10
95
1 -1 -1 6 -13 19 -21D1
y
a5b3a3b3
14n27434
6
10
87
1 -1 -1 6 -12 17 -19D1
y
a5b2a2bcb3c
15n5291
6
10
79
1 -1 -2 7 -11 14 -151
n
D
a4b2a3bcb2c2
15n5541
6
10
99
1 -1 -2 8 -14 18 -191
n
E
a4b2a3bcb3c
15n6098
6
10
99
1 -1 -2 8 -14 18 -191
n
E
a5b2a2bcb2c2
15n6237
6
10
79
1 -1 -2 7 -11 14 -151
n
D
a5ba2b2cb3c
15n11690
6
10
19
1 -1 -1 3 -3 3 -31
n
a3ba2b2c3b2c2
15n11787
6
10
31
1 -1 -2 6 -6 3 -11
n
F
a3ba2b2c2b2c3
15n11995
6
10
31
1 -1 -2 6 -6 3 -11
n
F
a3b4a2bcb3c
15n12185
6
10
115
1 -1 -2 8 -15 22 -25D1
y
G
a4ba2cb2ac2b2
15n12325
6
8
9
1 -1 -1 3 -1 -4 71
n
a3b4a2bcb2c2
15n13465
6
10
115
1 -1 -2 8 -15 22 -25Z2
n
G
a5b3a2cb3c
15n18803
6
10
159
1 -1 -3 12 -22 29 -31D1
y
H
a5b2a2cb3c2
15n21371
6
10
159
1 -1 -3 12 -22 29 -311
n
H
a3ba2b3c2bc3
15n22078
6
10
7
1 -1 -1 3 -2 0 1D1
y
a4ba3b2cb3c
15n22340
6
10
27
1 -1 -1 4 -5 4 -31
n
a3cb3cb2a2b3
15n23225
6
10
67
1 -1 -2 7 -10 11 -111
n
I
a3b3a3bcb3c
15n23383
6
10
123
1 -1 -2 9 -17 23 -251
n
a3ba2bc2b3c3
15n23603
6
10
67
1 -1 -2 7 -10 11 -111
n
I
a4b3a3cb3c
15n26942
6
10
195
1 -1 -3 13 -26 37 -41D1
y
J
a3b2acb2ac2b3
15n27229
6
8
29
1 -1 -1 2 2 -8 111
n
a3b3a2cb5c
15n27449
6
10
175
1 -1 -3 12 -23 33 -371
n
K
a3b2a2cb5c2
15n28185
6
10
175
1 -1 -3 12 -23 33 -371
n
K
a4b2a3cb3c2
15n28732
6
10
195
1 -1 -3 13 -26 37 -411
n
J
a3ba2b2c3bc3
15n30327
6
10
15
1 -1 -1 4 -4 1 1D1
y
a5ba3bcb3c
15n30419
6
10
43
1 -1 -1 4 -6 8 -9D1
y
L
a5ba3bcb2c2
15n30444
6
10
43
1 -1 -1 4 -6 8 -9D1
y
L
a6b3acb3c
15n31832
6
10
87
1 -1 -2 8 -13 15 -15D2
y
M
a2b2a2b3acb3c
15n32158
6
8
41
1 -1 -1 2 3 -11 151
n
a4b2acb5c2
15n32328
6
10
115
1 -1 -2 8 -15 22 -25D1
y
N
a6b2acb3c2
15n32537
6
10
87
1 -1 -2 8 -13 15 -15D1
y
M
a3ba3bcba3bc
15n40211
6
8
9
1 -1 0 0 1 -1 1D9
y
(2,9) cable of the trefoil
a3ba3bcba2bc2
15n41185
6
8
5
1 -1 0 0 1 0 -1D1
y
T(4,5)
a5ba2bcb3c2
15n41316
6
10
39
1 -1 -1 4 -6 7 -7D1
y
a3b2a2cb2a2bcb
15n43686
6
8
41
1 -1 0 -1 5 -8 91
n
a3ba3bc2b2c3
15n43833
6
10
51
1 -1 -2 7 -9 7 -51
n
a3ba3bcba2b2c
15n53717
6
8
17
1 -1 0 0 2 -3 3D1
y
a3b3a2cb3c3
15n53776
6
10
231
1 -1 -3 14 -30 45 -51D1
y
O
a3b3a2cb4c2
15n54096
6
10
231
1 -1 -3 14 -30 45 -511
n
O
a3b2acb3ac2b2
15n55590
6
8
21
1 -1 -1 3 0 -7 11D1
y
a4ba3bcb3c2
15n62359
6
10
51
1 -1 -1 5 -8 9 -9D1
y
a4b3acb3c3
15n62615
6
10
159
1 -1 -2 10 -21 31 -35D2
y
P
a4b3acb4c2
15n62682
6
10
159
1 -1 -2 10 -21 31 -35D1
y
P
a3b2acb2a3bcb
15n80794
6
8
33
1 -1 0 0 3 -7 9D1
y
a5b3acb3c2
15n85223
6
10
135
1 -1 -2 10 -19 25 -27D2
y
a3b3a3cb3c2
15n92754
6
10
243
1 -1 -3 15 -32 47 -53D2
y
a3ba3b2cb3c2
15n100284
6
10
27
1 -1 -1 5 -6 3 -1D4
y
a4b2ac2b3c3
15n115253
6
10
175
1 -1 -3 12 -23 33 -371
n
K
a4b2acb3c4
15n115416
6
10
115
1 -1 -2 8 -15 22 -25D2
y
N
a2b2c3ba3b2c2
15n117854
6
10
91
1 -1 -3 10 -14 14 -13D1
y
R
a3c2b2c3ba2b2
15n117860
6
10
91
1 -1 -3 10 -14 14 -131
n
R
15n1181696
10
235
1 -1 -4 16 -31 44 -491
n
S
a3b2a2c3b3c2
15n118265
6
10
235
1 -1 -4 16 -31 44 -49D1
y
S
a3b2a2c2b3c3
15n118706
6
10
235
1 -1 -4 16 -31 44 -49Z2
n
S
a3c3b2a2bc2b2
15n118779
6
10
91
1 -1 -3 10 -14 14 -13D1
y
R
a3b2c2ba2bc3b
15n119483
6
8
17
1 -1 -2 5 -1 -8 13D1
y
T
a3bc2b2a2bc3b
15n122907
6
8
17
1 -1 -2 5 -1 -8 13D3
y
T
a2b2a2cb2acb3c
15n129772
6
8
89
1 -1 0 -2 10 -19 231
n
a3b2a2bcba2b2c
15n140472
6
8
29
1 -1 0 -1 4 -5 5D1
y
a4b2acb2a2bcb
15n140504
6
8
25
1 -1 0 0 2 -5 7D1
y
a2b2acb2a2cb3c
15n143434
6
8
77
1 -1 0 -2 9 -16 19D1
y
a2b2acb2a2bcb2c
15n143746
6
8
45
1 -1 0 -2 7 -8 7D2
y
a2b2acb3acb3c
15n151082
6
8
81
1 -1 0 -1 8 -18 231
n
a3b2acb2acb3c
15n161040
6
8
53
1 -1 0 -1 6 -11 131
n
ab2acb2acb5c
15n163295
6
8
33
1 -1 0 0 3 -7 9D2
y
a3b2acb3acb2c
15n163311
6
8
65
1 -1 0 -1 7 -14 17D1
y
ab2acb3acb4c
15n163697
6
8
57
1 -1 0 0 5 -13 17Z2
n
ab3acb3acb3c
15n168025
6
8
81
1 -1 0 0 7 -19 25D6
y
a3ba2c3b2dc2d2
16n29507
6
10
171
1 -1 -4 15 -25 29 -291
n
U
a3ba2c3b2dc3d
16n32513
6
10
171
1 -1 -4 15 -25 29 -291
n
U
a3ba2c2b2dc3d2
16n35008
6
10
171
1 -1 -4 15 -25 29 -291
n
U
a3ba2bc2bdc3d2
16n60983
6
10
63
1 -1 -3 10 -12 7 -31
n
V
a2b2acb2dc3bdc2
16n87023
6
8
81
1 -1 -2 4 5 -23 33D1
y
W
a2b2ac2bdc2b2cdc
16n92582
6
8
65
1 -1 -1 1 6 -16 211
n
a2b2ac3b3dc3d
16n93564
6
10
315
1 -1 -5 21 -42 59 -65D1
y
X
a2b2acb2dc2bd2c2
16n102459
6
8
81
1 -1 -2 4 5 -23 33Z2
n
W
a2b2ac2b3dc3d2
16n108923
6
10
315
1 -1 -5 21 -42 59 -65D1
y
X
a2b2acbdc2b2d2c2
16n126724
6
8
45
1 -1 -2 5 1 -15 23D1
y
Y
a2b2acb2dc3bd2c
16n127191
6
8
45
1 -1 -2 5 1 -15 23D1
y
Y
a3ba2bc3bdc3d
16n129379
6
10
63
1 -1 -3 10 -12 7 -3D1
y
V
16n1449586
10
459
1 -1 -6 27 -59 89 -1011
n
AA
16n1495176
10
459
1 -1 -6 27 -59 89 -1011
n
AA
a3ba2c3bdc3d2
16n161647
6
10
135
1 -1 -3 12 -20 23 -23D1
y
a3ba3c3bdc3d
16n167518
6
10
99
1 -1 -3 11 -16 15 -13D2
y
Z
16n1738946
10
351
1 -1 -5 22 -46 67 -75D1
y
BB
a3ba3c2bdc3d2
16n174567
6
10
99
1 -1 -3 11 -16 15 -13D1
y
Z
16n1753246
10
351
1 -1 -5 22 -46 67 -75D1
y
BB
ab3ac3b3dc3d
16n179454
6
10
315
1 -1 -5 21 -42 59 -65D4
y
X
16n3396386
10
459
1 -1 -5 25 -58 91 -105D2
y
abac2b2acdcb2c2d
16n843750
6
8
81
1 -1 0 -3 11 -16 17D6
y
17 crossing knot with DT-code 4 8 -18 2 32 28 24 -20 -22 -6 -16 26 14 30 12 34 106
10
891
1 -1 -9 45 -110 179 -209D2
y

Legend for the columns

  1. Braid: A positive braid word whose closure is the knot. a, b, c, ... stand for the Artin generators σ1, σ2, σ3, .... Every braid word has length equal to the crossing number of the knot, and number of strands equal to the minimal number of strands among all positive braids whose closure is that knot.
  2. Number: The knot's number in the usual tables (for crossing number 10 or less, that is Rolfen's table, for crossing number between 11 and 16, the one by Hoste-Thistlethwaite-Weeks [3], which can be accessed via knotscape [4]).
  3. Brick diagram / Checkerboard graph: The brick diagram (in black) is just a simpler way to draw a positive braid diagram, replacing crossings by short vertical intervals. Bricks (i.e. innermost rectangles) correspond to Hopf bands in the canonical Seifert surface, whose core curves give a basis of the first homology group. The checkerboard graph (in blue) has a vertex per brick, and an edge for every pair of non-trivially intersecting homology classes. The edges come with an orientation, and the bounded regions with a checkerboard coloring. Edges and colorings are only drawn if they matter for the knot type. See [1] for details, where it is also shown that one may recover the knot from the checkerboard graph. For arborescent knots, the checkerboard graph corresponds to the trees considered by Bonahon-Siebenmann [2], up to some orientation issues.
  4. g: The genus of the knot, equal to half the number of vertices of the checkerboard graph.
  5. sig: The signature of the knot.
  6. det: The determinant of the knot.
  7. Alexander pol.: The coefficients of the Alexander polynomial of the knot. The Alexander polynomial is reciprocal, so to save space, only half of the coefficients are printed; e.g., "1 -1" is short for "1 - t + t2".
  8. Sym.: The symmetry group, as given by knotscape for hyperbolic knots, and listed in [3] for non-hyperbolic knots. The symmetry group is the subgroup of homeomorphisms of S3 mapping the (unoriented) knot to itself, modulo homeomorphisms isotopic to the identity.
  9. Inv.: y for invertible knots, n for non-invertible knots. A knot is invertible if it is isotopic to a copy of itself with reversed orientation.
  10. Mutants: Families of mutant knots are named A, B, ..., Z, AA, BB.
  11. Torus or Satellite: This column keeps track of non-hyperbolic knots.

How the list was compiled

First, I made a list of positive braid words containing at least one, but usually a lot of words for each positive braid knot of genus six or less. Then, I used knotscape [4] to identify the knots, and remove the duplicates.

For the first step, note that every positive braid knot is the closure of a positive braid word in which every generator appears at least twice. This gives a (non-optimal) upper bound on the number of strands one has to consider, and there are only finitely braid words respecting that bound for knots of fixed genus. The list can be made shorter by similar considerations (e.g. one can restrict oneself to braid words that are lexicographically minimal among their cyclcic conjugates). Here is the link to a short C++-program I wrote that prints the list: listbraids.

This program can also be used to make lists for higher genera. However, knotscape does not identify and enumerate diagrams with 17 or more crossings, so one would have to proceed with a different tool to remove duplicates from the list.

The list of arborescent knots was compiled by hand. They may also be identified with knotscape.

Other lists

  • Up to 12 crossings, the information agrees with that of KnotInfo [5].
  • Alexander Stoimenow made a list of positive braid knots with crossing number 16 or less for his paper [6], which agrees with this list.
  • Stoimenow also made a list of mutant knots with crossing number 15 or less [7], which agrees with the mutant families given above.

References

[1] Sebastian Baader, Lukas Lewark and Livio Liechti: Checkerboard graph monodromies, Preprint. arXiv:1706.09210 pdf.

[2] Francis Bonahon, Laurent C. Siebenmann: New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots, Preprint. http://www-bcf.usc.edu/~fbonahon/Research/Publications.html.

[3] Jim Hoste, Morwen Thistlethwaite and Jeff Weeks: The first 1,701,936 knots, Math. Intelligencer 20 (1998), no. 4, 33–48. MR1646740.

[4] Jim Hoste and Morwen Thistlethwaite: Knotscape, version 1.01, 1999. http://www.math.utk.edu/~morwen/knotscape.html.

[5] Jae Choon Cha and Charles Livingston: KnotInfo: Table of Knot Invariants, retrieved July 7, 2017. http://www.indiana.edu/~knotinfo.

[6] Alexander Stoimenow: On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks, Trans. Amer. Math. Soc. 354 (2002), no. 10, 3927–3954. MR1926860, arXiv:math/0110016 pdf

[6] Alexander Stoimenow: Knot data tables, retrieved July 7, 2017. http://stoimenov.net/stoimeno/homepage/ptab/.

Last update: October 1st, 2017.