# L10n63

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n63 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u v+1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1}$ (db) Kauffman polynomial $-3 a^2 z^8-3 z^8-6 a^3 z^7-12 a z^7-6 z^7 a^{-1} -3 a^4 z^6-a^2 z^6-4 z^6 a^{-2} -2 z^6+13 a^3 z^5+26 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^2 z^4+7 z^4 a^{-2} +13 z^4-6 a^5 z^3-18 a^3 z^3-18 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-2 z^2 a^{-2} -5 z^2+6 a^5 z+9 a^3 z+4 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-4-3-2-101234χ
8        11
6       3 -3
4      41 3
2     53  -2
0    64   2
-2   56    1
-4  45     -1
-6 25      3
-814       -3
-103        3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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