# L8a6

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a6 at Knotilus! L8a6 is $8^2_{6}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v-3 u-3 v+2}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-q^{7/2}+\frac{1}{q^{7/2}}+2 q^{5/2}-\frac{3}{q^{5/2}}-2 q^{3/2}+\frac{3}{q^{3/2}}+3 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} -2 a^3 z-a^3 z^{-1} -z a^{-3} +a z^3+z^3 a^{-1} +z a^{-1}$ (db) Kauffman polynomial $-a z^7-z^7 a^{-1} -a^2 z^6-2 z^6 a^{-2} -3 z^6-a^3 z^5+2 a z^5+2 z^5 a^{-1} -z^5 a^{-3} -a^4 z^4+7 z^4 a^{-2} +8 z^4-a^5 z^3-a^3 z^3-3 a z^3+3 z^3 a^{-3} -5 z^2 a^{-2} -5 z^2+2 a^5 z+2 a^3 z+a z-z a^{-3} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-4-3-2-101234χ
8        11
6       1 -1
4      11 0
2     21  -1
0    21   1
-2   23    1
-4  11     0
-6  2      2
-811       0
-101        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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