# L10a86

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a86 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^4-3 u^2 v^3+4 u^2 v^2-3 u^2 v+u^2-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u+v^4-3 v^3+4 v^2-3 v+1}{u v^2}$ (db) Jones polynomial $-9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-17 q^{5/2}+\frac{4}{q^{5/2}}+18 q^{3/2}-\frac{9}{q^{3/2}}-q^{13/2}+4 q^{11/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} +2 a z^3-7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +2 a z-5 z a^{-1} +5 z a^{-3} -z a^{-5} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $-3 z^9 a^{-1} -3 z^9 a^{-3} -16 z^8 a^{-2} -8 z^8 a^{-4} -8 z^8-8 a z^7-12 z^7 a^{-1} -12 z^7 a^{-3} -8 z^7 a^{-5} -4 a^2 z^6+30 z^6 a^{-2} +11 z^6 a^{-4} -4 z^6 a^{-6} +11 z^6-a^3 z^5+14 a z^5+36 z^5 a^{-1} +36 z^5 a^{-3} +14 z^5 a^{-5} -z^5 a^{-7} +5 a^2 z^4-18 z^4 a^{-2} -4 z^4 a^{-4} +5 z^4 a^{-6} -4 z^4+a^3 z^3-9 a z^3-31 z^3 a^{-1} -31 z^3 a^{-3} -9 z^3 a^{-5} +z^3 a^{-7} -a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2+4 a z+12 z a^{-1} +12 z a^{-3} +4 z a^{-5} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 \
-4-3-2-10123456χ
14          11
12         3 -3
10        61 5
8       73  -4
6      106   4
4     98    -1
2    89     -1
0   610      4
-2  37       -4
-4 16        5
-6 3         -3
-81          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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