# L9a42

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a42 at Knotilus! L9a42 is $9^2_{41}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(u^2 v^2-u v^2+3 u v-u+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-9 q^{5/2}+\frac{3}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-10 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-5 z^5 a^{-1} +z^5 a^{-3} +3 a z^3-9 z^3 a^{-1} +3 z^3 a^{-3} +3 a z-6 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +5 z^6 a^{-4} -6 z^4 a^{-4} +3 z^2 a^{-4} +5 z^7 a^{-3} +a^3 z^5-6 z^5 a^{-3} -2 a^3 z^3+4 z^3 a^{-3} +a^3 z-2 z a^{-3} +2 z^8 a^{-2} +3 a^2 z^6+5 z^6 a^{-2} -6 a^2 z^4-12 z^4 a^{-2} +3 a^2 z^2+6 z^2 a^{-2} +4 a z^7+9 z^7 a^{-1} -7 a z^5-17 z^5 a^{-1} +3 a z^3+12 z^3 a^{-1} -2 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8+3 z^6-11 z^4+5 z^2-1$ (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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       41 -3
6      52  3
4     44   0
2    65    1
0   46     2
-2  24      -2
-4 14       3
-6 2        -2
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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