# L9a8

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a8 at Knotilus! L9a8 is $9^2_{25}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-8 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-1} -2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-3 a z+3 z a^{-1} -2 z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -4 z^3 a^{-5} +2 z a^{-5} +4 z^6 a^{-4} -4 z^4 a^{-4} +z^2 a^{-4} +3 z^7 a^{-3} +a^3 z^5+z^5 a^{-3} -3 a^3 z^3-7 z^3 a^{-3} +3 a^3 z+5 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +z^8 a^{-2} +2 a^2 z^6+6 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +2 a^2 z^2+3 z^2 a^{-2} +2 a z^7+5 z^7 a^{-1} -3 z^5 a^{-1} -8 a z^3-8 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8+4 z^6-9 z^4+3 z^2-1$ (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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       31 -2
6      52  3
4     33   0
2    55    0
0   45     1
-2  13      -2
-4 14       3
-6 1        -1
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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