# L11a51

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a51 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^2+1\right) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{7/2}}$ (db) Jones polynomial $-q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-11 q^{5/2}+12 q^{3/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^9 a^{-1} -a z^7+7 z^7 a^{-1} -z^7 a^{-3} -5 a z^5+17 z^5 a^{-1} -5 z^5 a^{-3} -7 a z^3+16 z^3 a^{-1} -7 z^3 a^{-3} -2 a z+4 z a^{-1} -2 z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10}-4 a z^9-8 z^9 a^{-1} -4 z^9 a^{-3} -4 a^2 z^8+4 z^8 a^{-2} -4 z^8 a^{-4} +4 z^8-3 a^3 z^7+15 a z^7+36 z^7 a^{-1} +14 z^7 a^{-3} -4 z^7 a^{-5} -a^4 z^6+13 a^2 z^6+9 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6+10 a^3 z^5-21 a z^5-65 z^5 a^{-1} -24 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-9 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} +7 z^4 a^{-6} -10 z^4-6 a^3 z^3+16 a z^3+44 z^3 a^{-1} +16 z^3 a^{-3} -4 z^3 a^{-5} +2 z^3 a^{-7} -a^4 z^2+a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +5 z^2-4 a z-8 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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          2 -2
10         31 2
8        52  -3
6       63   3
4      65    -1
2     76     1
0    58      3
-2   35       -2
-4  25        3
-6 13         -2
-8 2          2
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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