# L9n10

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n10 at Knotilus! L9n10 is $9^2_{58}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^3-t(2)^3-2 t(1) t(2)^2+3 t(2)^2+3 t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{2}{q^{9/2}}+\frac{3}{q^{7/2}}+q^{5/2}-\frac{5}{q^{5/2}}-3 q^{3/2}+\frac{5}{q^{3/2}}+4 \sqrt{q}-\frac{5}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} +z^3 a^3-a^3 z^{-1} -z^5 a-3 z^3 a-3 z a+z^3 a^{-1} +z a^{-1}$ (db) Kauffman polynomial $3 a^5 z^3-4 a^5 z+a^5 z^{-1} +a^4 z^6+a^4 z^2-a^4+a^3 z^7+2 a^3 z^3-3 a^3 z+a^3 z^{-1} +4 a^2 z^6-5 a^2 z^4+z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} +a z^7+3 a z^5+3 z^5 a^{-1} -6 a z^3-5 z^3 a^{-1} +2 a z+z a^{-1} +3 z^6-4 z^4$ (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 \
-4-3-2-10123χ
6       1-1
4      2 2
2     21 -1
0    32  1
-2   33   0
-4  22    0
-6 13     2
-812      -1
-102       2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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