# L11a82

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a82 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^4-10 u v^3+14 u v^2-6 u v+u+v^5-6 v^4+14 v^3-10 v^2+2 v}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{22}{q^{5/2}}-8 q^{3/2}+\frac{21}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+14 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^7+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} -2 z^5 a^3-4 z^3 a^3-8 z a^3-4 a^3 z^{-1} -z^5 a+3 z^3 a+6 z a+3 a z^{-1} +2 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} -z a^{-3}$ (db) Kauffman polynomial $a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-10 a^7 z^5+8 a^7 z^3-3 a^7 z+6 a^6 z^8-12 a^6 z^6+5 a^6 z^4-a^6 z^2+a^6+4 a^5 z^9+5 a^5 z^7-34 a^5 z^5+36 a^5 z^3-15 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+17 a^4 z^8-47 a^4 z^6+40 a^4 z^4-14 a^4 z^2+2 a^4+9 a^3 z^9-a^3 z^7-40 a^3 z^5+z^5 a^{-3} +53 a^3 z^3-2 z^3 a^{-3} -24 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^{10}+19 a^2 z^8-51 a^2 z^6+3 z^6 a^{-2} +53 a^2 z^4-4 z^4 a^{-2} -24 a^2 z^2+z^2 a^{-2} +3 a^2+5 a z^9+4 a z^7+6 z^7 a^{-1} -26 a z^5-9 z^5 a^{-1} +34 a z^3+7 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +8 z^8-14 z^6+16 z^4-11 z^2+1$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         61 5
2        82  -6
0       106   4
-2      129    -3
-4     109     1
-6    812      4
-8   610       -4
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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