# L11a15

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a15 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-12 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+11 q^{3/2}-\frac{3}{q^{3/2}}-q^{17/2}+4 q^{15/2}-6 q^{13/2}+9 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-7} + a^{-7} z^{-1} +z^5 a^{-5} +z^3 a^{-5} -z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} -z a^{-3} +z^5 a^{-1} -a z^3+2 z^3 a^{-1} -a z+3 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^7 a^{-9} -3 z^5 a^{-9} +z^3 a^{-9} +4 z^8 a^{-8} -16 z^6 a^{-8} +15 z^4 a^{-8} -z^2 a^{-8} -2 a^{-8} +5 z^9 a^{-7} -20 z^7 a^{-7} +21 z^5 a^{-7} -6 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} -21 z^6 a^{-6} +24 z^4 a^{-6} +z^2 a^{-6} -5 a^{-6} +9 z^9 a^{-5} -33 z^7 a^{-5} +36 z^5 a^{-5} -13 z^3 a^{-5} -z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} -11 z^6 a^{-4} +9 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +4 z^9 a^{-3} -8 z^7 a^{-3} +9 z^5 a^{-3} -9 z^3 a^{-3} +2 z a^{-3} +4 z^8 a^{-2} -2 z^6 a^{-2} +a^2 z^4-4 z^4 a^{-2} -a^2 z^2+z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} +3 a z^5-4 a z^3-7 z^3 a^{-1} +2 a z+5 z a^{-1} - a^{-1} z^{-1} +4 z^6-3 z^4$ (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          3 -3
14         31 2
12        63  -3
10       63   3
8      76    -1
6     76     1
4    47      3
2   57       -2
0  26        4
-2 13         -2
-4 2          2
-61           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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