# L11n170

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n170 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v+3 u v^3-5 u v^2+3 u v-v^3+3 v^2-3 v+1}{u v^2}$ (db) Jones polynomial $-7 q^{9/2}+9 q^{7/2}-10 q^{5/2}+7 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-5} +3 z^3 a^{-5} +3 z a^{-5} +2 a^{-5} z^{-1} -z^7 a^{-3} -5 z^5 a^{-3} -9 z^3 a^{-3} -8 z a^{-3} -5 a^{-3} z^{-1} +z^5 a^{-1} +3 z^3 a^{-1} +4 z a^{-1} +3 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -4 z^8 a^{-4} -3 z^8 a^{-6} -3 z^7 a^{-5} -3 z^7 a^{-7} -2 z^6 a^{-2} +5 z^6 a^{-4} +6 z^6 a^{-6} -z^6 a^{-8} -7 z^5 a^{-1} -5 z^5 a^{-3} +11 z^5 a^{-5} +9 z^5 a^{-7} +3 z^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} -3 z^4-a z^3+9 z^3 a^{-1} +14 z^3 a^{-3} -2 z^3 a^{-5} -6 z^3 a^{-7} +5 z^2 a^{-2} +2 z^2 a^{-4} -6 z^2 a^{-6} -3 z^2 a^{-8} -7 z a^{-1} -10 z a^{-3} -3 z a^{-5} -5 a^{-2} -5 a^{-4} + a^{-8} +3 a^{-1} z^{-1} +5 a^{-3} z^{-1} +2 a^{-5} z^{-1}$ (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 \
-2-101234567χ
16         1-1
14        2 2
12       41 -3
10      32  1
8     64   -2
6    43    1
4   36     3
2  44      0
0  4       4
-213        -2
-41         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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