# L11n182

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n182 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^3-3 u^2 v^2+3 u^2 v+2 u v^4-5 u v^3+7 u v^2-5 u v+2 u+3 v^3-3 v^2+v}{u v^2}$ (db) Jones polynomial $2 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-a^3 z^3-2 a z^5-6 a z^3+2 z^3 a^{-1} -7 a z-2 a z^{-1} +3 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $a^7 z^5-2 a^7 z^3+3 a^6 z^6-5 a^6 z^4+6 a^5 z^7-16 a^5 z^5+16 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-10 a^4 z^6+6 a^4 z^4+a^4 z^2-a^4+2 a^3 z^9+2 a^3 z^7-10 a^3 z^5+10 a^3 z^3-2 a^3 z+8 a^2 z^8-21 a^2 z^6+27 a^2 z^4+3 z^4 a^{-2} -14 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +2 a z^9-3 a z^7+z^7 a^{-1} +10 a z^5+3 z^5 a^{-1} -15 a z^3-7 z^3 a^{-1} +9 a z+3 z a^{-1} -2 a z^{-1} - a^{-1} z^{-1} +3 z^8-8 z^6+19 z^4-20 z^2+5$ (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       52 -3
0      63  3
-2     76   -1
-4    55    0
-6   47     3
-8  35      -2
-10  4       4
-1213        -2
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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