# L11n51

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n51 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^4-2 v^3-2 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+8 q^{5/2}-8 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-8 z^3 a^{-1} +9 z^3 a^{-3} -4 z^3 a^{-5} +3 a z-9 z a^{-1} +9 z a^{-3} -4 z a^{-5} +z a^{-7} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $z^2 a^{-8} +4 z^3 a^{-7} -z a^{-7} +2 z^6 a^{-6} -z^4 a^{-6} +z^2 a^{-6} - a^{-6} +5 z^7 a^{-5} -15 z^5 a^{-5} +16 z^3 a^{-5} -6 z a^{-5} + a^{-5} z^{-1} +4 z^8 a^{-4} -10 z^6 a^{-4} +3 z^4 a^{-4} +4 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +6 z^7 a^{-3} -32 z^5 a^{-3} +37 z^3 a^{-3} -17 z a^{-3} +3 a^{-3} z^{-1} +6 z^8 a^{-2} -20 z^6 a^{-2} +12 z^4 a^{-2} +4 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+2 z^7 a^{-1} -5 a z^5-22 z^5 a^{-1} +9 a z^3+34 z^3 a^{-1} -7 a z-19 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4-2$ (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 \
-4-3-2-1012345χ
14         1-1
12        3 3
10       31 -2
8      43  1
6     43   -1
4    44    0
2   46     2
0  12      -1
-2 14       3
-4 1        -1
-61         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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