# L11n316

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n316 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^2 w-u v^2+u v w^2-3 u v w+2 u v-u w^2+u w-v^2 w+v^2-2 v w^2+3 v w-v+w^2-2 w}{\sqrt{u} v w}$ (db) Jones polynomial $q^9-3 q^8+5 q^7-6 q^6+8 q^5-7 q^4+7 q^3-4 q^2+3 q$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-8} -z^4 a^{-6} + a^{-6} z^{-2} +2 a^{-6} -2 z^4 a^{-4} -5 z^2 a^{-4} -2 a^{-4} z^{-2} -6 a^{-4} +3 z^2 a^{-2} + a^{-2} z^{-2} +4 a^{-2}$ (db) Kauffman polynomial $z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -2 z^7 a^{-5} +z^7 a^{-7} +3 z^7 a^{-9} -z^6 a^{-4} -11 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-3} +5 z^5 a^{-5} -8 z^5 a^{-7} -10 z^5 a^{-9} -z^4 a^{-4} +8 z^4 a^{-6} +6 z^4 a^{-8} -3 z^4 a^{-10} -4 z^3 a^{-3} -8 z^3 a^{-5} +3 z^3 a^{-7} +7 z^3 a^{-9} +5 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-8} +2 z^2 a^{-10} +6 z a^{-3} +6 z a^{-5} -5 a^{-2} -8 a^{-4} -3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (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 \
012345678χ
19        11
17       2 -2
15      31 2
13     43  -1
11    42   2
9   34    1
7  44     0
5 14      3
323       -1
13        3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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