# L11n230

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n230 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^2-u^3 v+2 u^2 v^3-4 u^2 v^2+5 u^2 v-2 u^2-2 u v^3+5 u v^2-4 u v+2 u-v^2+v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $2 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^5-2 a^3 z^3-2 a^3 z-a^3 z^{-1} -2 a z^5-6 a z^3+2 z^3 a^{-1} -5 a z+3 z a^{-1}$ (db) Kauffman polynomial $a^7 z^5-3 a^7 z^3+2 a^7 z+2 a^6 z^6-4 a^6 z^4+a^6 z^2+3 a^5 z^7-6 a^5 z^5+5 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-6 a^4 z^6+7 a^4 z^4-3 a^4 z^2-a^4+a^3 z^9+4 a^3 z^7-13 a^3 z^5+16 a^3 z^3-7 a^3 z+a^3 z^{-1} +5 a^2 z^8-10 a^2 z^6+11 a^2 z^4+3 z^4 a^{-2} -4 a^2 z^2-4 z^2 a^{-2} +a z^9+2 a z^7+z^7 a^{-1} -3 a z^5+3 z^5 a^{-1} -8 z^3 a^{-1} +2 a z+3 z a^{-1} +2 z^8-2 z^6+3 z^4-4 z^2$ (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 \
-6-5-4-3-2-10123χ
6         2-2
4        3 3
2       42 -2
0      63  3
-2     55   0
-4    55    0
-6   35     2
-8  25      -3
-10 14       3
-12 1        -1
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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