# L11a359

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a359 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(t(2) t(1)^2-t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)+t(2)\right) \left(t(2) t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)-t(2)^2+t(2)\right)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-q^{13/2}+3 q^{11/2}-6 q^{9/2}+10 q^{7/2}-14 q^{5/2}+15 q^{3/2}-16 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} -z a^{-5} +z^5 a^{-3} -a^3 z^3+z^3 a^{-3} -a^3 z- a^{-3} z^{-1} +a z^5+2 z^5 a^{-1} +a z^3+4 z^3 a^{-1} +3 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -6 z^4 a^{-6} +2 z^2 a^{-6} +5 z^7 a^{-5} -11 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +5 z^8 a^{-4} +a^4 z^6-9 z^6 a^{-4} -3 a^4 z^4+7 z^4 a^{-4} +2 a^4 z^2-z^2 a^{-4} +3 z^9 a^{-3} +3 a^3 z^7-9 a^3 z^5-7 z^5 a^{-3} +7 a^3 z^3+9 z^3 a^{-3} -a^3 z-z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +4 a^2 z^8+6 z^8 a^{-2} -10 a^2 z^6-15 z^6 a^{-2} +6 a^2 z^4+15 z^4 a^{-2} -a^2 z^2-5 z^2 a^{-2} + a^{-2} +3 a z^9+6 z^9 a^{-1} -4 a z^7-12 z^7 a^{-1} +14 z^5 a^{-1} -3 a z^3-12 z^3 a^{-1} +2 a z+5 z a^{-1} - a^{-1} z^{-1} +z^{10}+5 z^8-14 z^6+11 z^4-5 z^2$ (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 \
-5-4-3-2-10123456χ
14           11
12          2 -2
10         41 3
8        62  -4
6       84   4
4      87    -1
2     87     1
0    69      3
-2   47       -3
-4  26        4
-6 14         -3
-8 2          2
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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