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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a249's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,8,17,7 X18,12,19,11 X2,19,3,20 X12,4,13,3 X20,13,21,14 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21
Gauss code {1, -4, 5, -11, 7, -8, 2, -10}, {8, -1, 3, -5, 6, -7, 9, -2, 10, -3, 4, -6, 11, -9}
A Braid Representative
A Morse Link Presentation L11a249 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^4-2 u^2 v^3+u^2 v^2-u v^4+3 u v^3-6 u v^2+3 u v-u+v^2-2 v+1\right)}{u^{3/2} v^{5/2}} (db)
Jones polynomial -q^{13/2}+5 q^{11/2}-11 q^{9/2}+18 q^{7/2}-25 q^{5/2}+28 q^{3/2}-29 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^9 a^{-1} -a z^7+5 z^7 a^{-1} -z^7 a^{-3} -3 a z^5+7 z^5 a^{-1} -3 z^5 a^{-3} -a z^3-z^3 a^{-3} +2 a z-4 z a^{-1} +2 z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} +5 z^6 a^{-6} -4 z^4 a^{-6} +11 z^7 a^{-5} -14 z^5 a^{-5} +4 z^3 a^{-5} +14 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -a^4 z^4+4 z^4 a^{-4} +z^2 a^{-4} +11 z^9 a^{-3} +5 a^3 z^7-6 z^7 a^{-3} -9 a^3 z^5-14 z^5 a^{-3} +3 a^3 z^3+13 z^3 a^{-3} -4 z a^{-3} +4 z^{10} a^{-2} +10 a^2 z^8+18 z^8 a^{-2} -21 a^2 z^6-48 z^6 a^{-2} +11 a^2 z^4+30 z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +10 a z^9+21 z^9 a^{-1} -15 a z^7-37 z^7 a^{-1} -2 a z^5+8 z^5 a^{-1} +9 a z^3+15 z^3 a^{-1} -4 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +4 z^{10}+14 z^8-47 z^6+34 z^4-5 z^2-1 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
14           11
12          4 -4
10         71 6
8        114  -7
6       147   7
4      1411    -3
2     1514     1
0    1116      5
-2   713       -6
-4  411        7
-6 17         -6
-8 4          4
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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