# L9a19

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a19 at Knotilus! L9a19 is $9^2_{38}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^3-5 u v^2+6 u v-2 u-2 v^3+6 v^2-5 v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-2 a^3 z^3-3 a^3 z-a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1}$ (db) Kauffman polynomial $a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+3 a^6 z^2+4 a^5 z^7-7 a^5 z^5+4 a^5 z^3-2 a^5 z+a^5 z^{-1} +2 a^4 z^8+4 a^4 z^6-13 a^4 z^4+7 a^4 z^2-a^4+10 a^3 z^7-18 a^3 z^5+10 a^3 z^3-3 a^3 z+a^3 z^{-1} +2 a^2 z^8+8 a^2 z^6-17 a^2 z^4+z^4 a^{-2} +6 a^2 z^2+6 a z^7-6 a z^5+4 z^5 a^{-1} +a z^3-3 z^3 a^{-1} +7 z^6-9 z^4+2 z^2$ (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 \
-6-5-4-3-2-10123χ
6         1-1
4        3 3
2       41 -3
0      53  2
-2     65   -1
-4    54    1
-6   36     3
-8  35      -2
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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