# L9a31

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a31 at Knotilus! L9a31 is $9^2_{39}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^2}$ (db) Jones polynomial $-q^{7/2}+3 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +9 a z^3-3 z^3 a^{-1} +7 a z-3 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4-a^6 z^2+3 a^5 z^5-3 a^5 z^3+5 a^4 z^6-7 a^4 z^4+4 a^4 z^2-a^4+5 a^3 z^7-8 a^3 z^5+z^5 a^{-3} +8 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +a^3 z^{-1} +2 a^2 z^8+4 a^2 z^6+3 z^6 a^{-2} -12 a^2 z^4-6 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+9 a z^7+4 z^7 a^{-1} -20 a z^5-8 z^5 a^{-1} +18 a z^3+5 z^3 a^{-1} -9 a z-4 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^8+2 z^6-10 z^4+7 z^2-3$ (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
8         11
6        2 -2
4       41 3
2      43  -1
0     63   3
-2    45    1
-4   45     -1
-6  24      2
-8 14       -3
-10 2        2
-121         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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