# L11a262

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a262 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-8 q^{9/2}+\frac{1}{q^{9/2}}+12 q^{7/2}-\frac{3}{q^{7/2}}-16 q^{5/2}+\frac{6}{q^{5/2}}+18 q^{3/2}-\frac{11}{q^{3/2}}-q^{13/2}+4 q^{11/2}-18 \sqrt{q}+\frac{14}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} +z^5 a^{-3} -a^3 z^3-a^3 z-z a^{-3} +a z^5+2 z^5 a^{-1} +a z^3+3 z^3 a^{-1} +a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +7 z^7 a^{-5} -12 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +7 z^8 a^{-4} +a^4 z^6-10 z^6 a^{-4} -3 a^4 z^4+4 z^4 a^{-4} +2 a^4 z^2-z^2 a^{-4} +4 z^9 a^{-3} +3 a^3 z^7+2 z^7 a^{-3} -9 a^3 z^5-13 z^5 a^{-3} +8 a^3 z^3+10 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +z^{10} a^{-2} +4 a^2 z^8+11 z^8 a^{-2} -9 a^2 z^6-27 z^6 a^{-2} +4 a^2 z^4+23 z^4 a^{-2} -5 z^2 a^{-2} +3 a z^9+7 z^9 a^{-1} -a z^7-9 z^7 a^{-1} -11 a z^5-2 z^5 a^{-1} +13 a z^3+9 z^3 a^{-1} -6 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+8 z^8-23 z^6+20 z^4-5 z^2-1$ (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 \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         51 4
8        73  -4
6       95   4
4      97    -2
2     99     0
0    711      4
-2   47       -3
-4  27        5
-6 14         -3
-8 2          2
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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