# L10a89

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a89 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v-u v^2-u-v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+7 q^{5/2}-7 q^{3/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +5 z a^{-3} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-7 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +4 z^6 a^{-6} -2 z^4 a^{-6} -3 z^2 a^{-6} +4 z^7 a^{-5} -5 z^5 a^{-5} -2 z^3 a^{-5} +2 z a^{-5} +3 z^8 a^{-4} -5 z^6 a^{-4} -z^4 a^{-4} +z^2 a^{-4} +z^9 a^{-3} +3 z^7 a^{-3} -18 z^5 a^{-3} +17 z^3 a^{-3} -4 z a^{-3} +5 z^8 a^{-2} -18 z^6 a^{-2} +15 z^4 a^{-2} -z^2 a^{-2} +z^9 a^{-1} +a z^7-5 a z^5-14 z^5 a^{-1} +8 a z^3+23 z^3 a^{-1} -5 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8-9 z^6+11 z^4-3 z^2-1$ (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 \
-4-3-2-10123456χ
16          11
14         2 -2
12        21 1
10       42  -2
8      42   2
6     34    1
4    44     0
2   35      2
0  12       -1
-2 13        2
-4 1         -1
-61          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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