# L9a3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a3 at Knotilus! L9a3 is $9^2_{33}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-6 q^{9/2}+8 q^{7/2}-10 q^{5/2}+\frac{1}{q^{5/2}}+9 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-1} +z^5 a^{-3} -a z^3+2 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a z+3 z a^{-1} -2 z a^{-3} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (db) Kauffman polynomial $-2 z^8 a^{-2} -2 z^8 a^{-4} -4 z^7 a^{-1} -9 z^7 a^{-3} -5 z^7 a^{-5} -4 z^6 a^{-2} -4 z^6 a^{-4} -4 z^6 a^{-6} -4 z^6-3 a z^5+2 z^5 a^{-1} +16 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+9 z^4 a^{-2} +13 z^4 a^{-4} +8 z^4 a^{-6} +3 z^4+4 a z^3+3 z^3 a^{-1} -6 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-2 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2 a^{-6} -2 a z-5 z a^{-1} -3 z a^{-3} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (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 \
-3-2-10123456χ
14         11
12        3 -3
10       31 2
8      53  -2
6     53   2
4    45    1
2   55     0
0  26      4
-2 13       -2
-4 2        2
-61         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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