# L9a10

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a10 at Knotilus! L9a10 is $9^2_{36}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-q^{9/2}+\frac{1}{q^{9/2}}+3 q^{7/2}-\frac{3}{q^{7/2}}-6 q^{5/2}+\frac{4}{q^{5/2}}+7 q^{3/2}-\frac{7}{q^{3/2}}-8 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a z^5+z^5 a^{-1} -a^3 z^3+2 a z^3+2 z^3 a^{-1} -z^3 a^{-3} -a^3 z+2 z a^{-1} -z a^{-3} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-5} +a^4 z^6-3 a^4 z^4+3 z^4 a^{-4} +2 a^4 z^2+3 a^3 z^7-11 a^3 z^5+6 z^5 a^{-3} +11 a^3 z^3-6 z^3 a^{-3} -a^3 z+3 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^8-3 a^2 z^6+7 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +4 a^2 z^2+4 z^2 a^{-2} +8 a z^7+5 z^7 a^{-1} -22 a z^5-5 z^5 a^{-1} +13 a z^3-5 z^3 a^{-1} +2 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^8+3 z^6-14 z^4+6 z^2-1$ (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 \
-5-4-3-2-101234χ
10         11
8        2 -2
6       41 3
4      32  -1
2     54   1
0    55    0
-2   23     -1
-4  25      3
-6 12       -1
-8 2        2
-101         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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