# L11a287

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a287 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u v+1) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $14 q^{9/2}-16 q^{7/2}+15 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+6 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-5} +5 z^5 a^{-5} +8 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} -z^9 a^{-3} -7 z^7 a^{-3} -18 z^5 a^{-3} -19 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +z^7 a^{-1} +5 z^5 a^{-1} +8 z^3 a^{-1} +4 z a^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10} a^{-4} -4 z^9 a^{-1} -9 z^9 a^{-3} -5 z^9 a^{-5} +z^8 a^{-2} -3 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+15 z^7 a^{-1} +29 z^7 a^{-3} +6 z^7 a^{-5} -7 z^7 a^{-7} +11 z^6 a^{-2} +14 z^6 a^{-4} +10 z^6 a^{-6} -5 z^6 a^{-8} +12 z^6+4 a z^5-19 z^5 a^{-1} -35 z^5 a^{-3} +9 z^5 a^{-7} -3 z^5 a^{-9} -12 z^4 a^{-2} -7 z^4 a^{-4} -4 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} -14 z^4-4 a z^3+13 z^3 a^{-1} +26 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +4 z^2 a^{-2} -2 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +4 z^2-5 z a^{-1} -5 z a^{-3} -z a^{-7} -z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         41 -3
12        72  5
10       85   -3
8      86    2
6     78     1
4    78      -1
2   48       4
0  26        -4
-2 14         3
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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