# L9a40

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a40 at Knotilus! L9a40 is $9^2_{4}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v+u v^2+u v+u+v\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^5-2 z a^5+z^5 a^3+3 z^3 a^3+z a^3+z^5 a+4 z^3 a+4 z a+a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^2+2 a^7 z^3+3 a^6 z^4-2 a^6 z^2+4 a^5 z^5-8 a^5 z^3+4 a^5 z+3 a^4 z^6-6 a^4 z^4+a^4 z^2+2 a^3 z^7-4 a^3 z^5-2 a^3 z^3+2 a^3 z+a^2 z^8-2 a^2 z^6-2 a^2 z^4+2 a^2 z^2+3 a z^7+z^7 a^{-1} -14 a z^5-6 z^5 a^{-1} +19 a z^3+11 z^3 a^{-1} -8 a z+a z^{-1} -6 z a^{-1} + a^{-1} z^{-1} +z^8-5 z^6+7 z^4-2 z^2-1$ (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
6         11
4          0
2       21 1
0      1   -1
-2     32   1
-4    23    1
-6   21     1
-8  12      1
-10 12       -1
-12 1        1
-141         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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