# L10a80

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a80 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^2 v^3-4 u^2 v^2+3 u^2 v-u^2+u v^4-4 u v^3+5 u v^2-4 u v+u-v^4+3 v^3-4 v^2+2 v}{u v^2}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-6 q^{5/2}+9 q^{3/2}-11 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -3 a^3 z^3-z^3 a^{-3} -6 a^3 z-2 a^3 z^{-1} -z a^{-3} +2 a z^5+z^5 a^{-1} +6 a z^3+z^3 a^{-1} +6 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^3 z^9-2 a z^9-3 a^4 z^8-10 a^2 z^8-7 z^8-a^5 z^7+a^3 z^7-8 a z^7-10 z^7 a^{-1} +12 a^4 z^6+34 a^2 z^6-9 z^6 a^{-2} +13 z^6+4 a^5 z^5+19 a^3 z^5+42 a z^5+21 z^5 a^{-1} -6 z^5 a^{-3} -14 a^4 z^4-30 a^2 z^4+13 z^4 a^{-2} -3 z^4 a^{-4} -6 a^5 z^3-32 a^3 z^3-44 a z^3-13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +4 a^4 z^2+7 a^2 z^2-5 z^2 a^{-2} -2 z^2+4 a^5 z+15 a^3 z+17 a z+5 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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       52  -3
2      64   2
0     66    0
-2    55     0
-4   47      3
-6  24       -2
-8 14        3
-10 2         -2
-121          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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