# L8a2

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a2 at Knotilus! L8a2 is $8^2_{10}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-5 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-1} -2 a z^3+3 z^3 a^{-1} -z^3 a^{-3} +a^3 z-4 a z+4 z a^{-1} -z a^{-3} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-5} +3 z^4 a^{-4} -z^2 a^{-4} +a^3 z^5+5 z^5 a^{-3} -3 a^3 z^3-6 z^3 a^{-3} +3 a^3 z+3 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^6+4 z^6 a^{-2} -5 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +a z^7+z^7 a^{-1} +3 a z^5+7 z^5 a^{-1} -13 a z^3-17 z^3 a^{-1} +10 a z+10 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+5 z^2-1$ (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
10        11
8       2 -2
6      31 2
4     22  0
2    43   1
0   34    1
-2  12     -1
-4 13      2
-6 1       -1
-81        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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