# L11a9

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a9 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(4 v^2-7 v+4\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-16 q^{9/2}+18 q^{7/2}-20 q^{5/2}+\frac{1}{q^{5/2}}+17 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-6 q^{13/2}+12 q^{11/2}-14 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-7} -z a^{-7} +z^5 a^{-5} +z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} +2 z^5 a^{-3} +2 z^3 a^{-3} -2 z a^{-3} -3 a^{-3} z^{-1} +z^5 a^{-1} -a z^3+z a^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^7 a^{-9} -4 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -12 z^6 a^{-8} +16 z^4 a^{-8} -8 z^2 a^{-8} +4 z^9 a^{-7} -13 z^7 a^{-7} +13 z^5 a^{-7} -6 z^3 a^{-7} +2 z a^{-7} +2 z^{10} a^{-6} +3 z^8 a^{-6} -26 z^6 a^{-6} +34 z^4 a^{-6} -17 z^2 a^{-6} + a^{-6} +11 z^9 a^{-5} -29 z^7 a^{-5} +20 z^5 a^{-5} -7 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +2 z^{10} a^{-4} +12 z^8 a^{-4} -42 z^6 a^{-4} +36 z^4 a^{-4} -13 z^2 a^{-4} +3 a^{-4} +7 z^9 a^{-3} -3 z^7 a^{-3} -18 z^5 a^{-3} +13 z^3 a^{-3} +3 z a^{-3} -3 a^{-3} z^{-1} +12 z^8 a^{-2} -20 z^6 a^{-2} +a^2 z^4+11 z^4 a^{-2} -4 z^2 a^{-2} +3 a^{-2} +12 z^7 a^{-1} +4 a z^5-17 z^5 a^{-1} -2 a z^3+8 z^3 a^{-1} +z a^{-1} -2 a^{-1} z^{-1} +8 z^6-6 z^4$ (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 \
-3-2-1012345678χ
18           11
16          2 -2
14         41 3
12        82  -6
10       84   4
8      108    -2
6     108     2
4    710      3
2   710       -3
0  39        6
-2 15         -4
-4 3          3
-61           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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