# L11a131

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a131 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(3 v^2-2 v+3\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $q^{21/2}-3 q^{19/2}+5 q^{17/2}-7 q^{15/2}+9 q^{13/2}-10 q^{11/2}+10 q^{9/2}-8 q^{7/2}+5 q^{5/2}-4 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} +z a^{-9} -z^5 a^{-7} -2 z^3 a^{-7} -z a^{-7} -z^5 a^{-5} -z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -3 z^3 a^{-3} -4 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-6} -z^{10} a^{-8} -z^9 a^{-5} -4 z^9 a^{-7} -3 z^9 a^{-9} -z^8 a^{-4} +3 z^8 a^{-6} -4 z^8 a^{-10} -z^7 a^{-3} +z^7 a^{-5} +15 z^7 a^{-7} +10 z^7 a^{-9} -3 z^7 a^{-11} -z^6 a^{-2} -8 z^6 a^{-6} +6 z^6 a^{-8} +14 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-5} -24 z^5 a^{-7} -11 z^5 a^{-9} +10 z^5 a^{-11} +z^4 a^{-2} -z^4 a^{-4} +6 z^4 a^{-6} -7 z^4 a^{-8} -12 z^4 a^{-10} +3 z^4 a^{-12} +4 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} +13 z^3 a^{-7} +7 z^3 a^{-9} -6 z^3 a^{-11} +3 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} -5 z a^{-1} -7 z a^{-3} -2 z a^{-5} -2 z a^{-7} -2 z a^{-9} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (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 \
-2-10123456789χ
22           1-1
20          2 2
18         31 -2
16        42  2
14       53   -2
12      54    1
10     55     0
8    35      -2
6   25       3
4  23        -1
2 14         3
0            0
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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