# L11n159

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n159 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^4-2 u^2 v^3+u^2 v^2-u v^4+u v^2-u+v^2-2 v+1}{u v^2}$ (db) Jones polynomial $2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $-z^7 a^{-5} +z^5 a^{-3} -5 z^5 a^{-5} +z^5 a^{-7} +4 z^3 a^{-3} -6 z^3 a^{-5} +4 z^3 a^{-7} +3 z a^{-3} -2 z a^{-5} +z a^{-7} -z a^{-9} + a^{-3} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-5} -z^9 a^{-7} -2 z^8 a^{-4} -3 z^8 a^{-6} -z^8 a^{-8} -z^7 a^{-3} +3 z^7 a^{-5} +4 z^7 a^{-7} +10 z^6 a^{-4} +15 z^6 a^{-6} +5 z^6 a^{-8} +5 z^5 a^{-3} +4 z^5 a^{-5} -z^5 a^{-7} -12 z^4 a^{-4} -19 z^4 a^{-6} -7 z^4 a^{-8} -7 z^3 a^{-3} -12 z^3 a^{-5} -5 z^3 a^{-7} +2 z^2 a^{-4} +8 z^2 a^{-6} +7 z^2 a^{-8} +z^2 a^{-10} +4 z a^{-3} +6 z a^{-5} +3 z a^{-7} +z a^{-9} + a^{-4} -3 a^{-6} -5 a^{-8} -2 a^{-10} - a^{-3} z^{-1} +2 a^{-7} z^{-1} + a^{-9} z^{-1}$ (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 \
-2-1012345χ
16       2-2
14      110
12     22 0
10    211 2
8   13   2
6  221   1
4 12     1
2 1      -1
01       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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