# L10a31

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a31 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-9 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^5+a^5 z^{-1} -2 z^3 a^3-4 z a^3-2 a^3 z^{-1} +z^5 a+2 z^3 a+2 z a+a z^{-1} +z^5 a^{-1} +2 z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-2 a^4 z^8-6 a^2 z^8-4 z^8-a^5 z^7-a^3 z^7-6 a z^7-6 z^7 a^{-1} +9 a^4 z^6+22 a^2 z^6-7 z^6 a^{-2} +6 z^6+5 a^5 z^5+18 a^3 z^5+27 a z^5+8 z^5 a^{-1} -6 z^5 a^{-3} -12 a^4 z^4-24 a^2 z^4+9 z^4 a^{-2} -3 z^4 a^{-4} -8 a^5 z^3-30 a^3 z^3-27 a z^3+2 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +5 a^4 z^2+11 a^2 z^2-2 z^2 a^{-2} +4 z^2+5 a^5 z+15 a^3 z+10 a z-3 z a^{-1} -3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-6-5-4-3-2-101234χ
10          11
8         2 -2
6        41 3
4       32  -1
2      64   2
0     55    0
-2    34     -1
-4   35      2
-6  13       -2
-8 13        2
-10 1         -1
-121          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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