# L8a9

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a9 at Knotilus! L8a9 is $8^2_{8}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1}{t(1) t(2)}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}+\frac{4}{q^{7/2}}+q^{5/2}-\frac{6}{q^{5/2}}-3 q^{3/2}+\frac{6}{q^{3/2}}+\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 (-z)+2 a^3 z^3+3 a^3 z+a^3 z^{-1} -a z^5-3 a z^3+z^3 a^{-1} -4 a z-a z^{-1} +z a^{-1}$ (db) Kauffman polynomial $a^6 z^4-a^6 z^2+3 a^5 z^5-5 a^5 z^3+2 a^5 z+3 a^4 z^6-3 a^4 z^4+a^3 z^7+6 a^3 z^5-13 a^3 z^3+7 a^3 z-a^3 z^{-1} +6 a^2 z^6-8 a^2 z^4+z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} +a^2+a z^7+6 a z^5+3 z^5 a^{-1} -13 a z^3-5 z^3 a^{-1} +7 a z+2 z a^{-1} -a z^{-1} +3 z^6-3 z^4$ (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 \
-5-4-3-2-10123χ
6        1-1
4       2 2
2      21 -1
0     42  2
-2    33   0
-4   33    0
-6  24     2
-8 12      -1
-10 2       2
-121        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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