# L10a90

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a90 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1)^2 (t(2)-1)^2 (t(1) t(2)+1)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{1}{q^{9/2}}+5 q^{7/2}-\frac{3}{q^{7/2}}-8 q^{5/2}+\frac{5}{q^{5/2}}+10 q^{3/2}-\frac{8}{q^{3/2}}+q^{11/2}-11 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -2 a^3 z+2 z a^{-3} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +7 a z^3-9 z^3 a^{-1} +6 a z-6 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-5 a z^7-6 z^7 a^{-1} -4 z^7 a^{-3} -a^4 z^6+7 a^2 z^6+z^6 a^{-2} -4 z^6 a^{-4} +13 z^6+10 a^3 z^5+22 a z^5+18 z^5 a^{-1} +3 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4-a^2 z^4+3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -6 z^4-9 a^3 z^3-22 a z^3-17 z^3 a^{-1} +4 z^3 a^{-5} -2 a^4 z^2-2 a^2 z^2-3 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -z^2+3 a^3 z+8 a z+6 z a^{-1} -z a^{-5} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-1012345χ
12          1-1
10         2 2
8        31 -2
6       52  3
4      53   -2
2     65    1
0    57     2
-2   34      -1
-4  25       3
-6 13        -2
-8 2         2
-101          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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