# L9a23

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a23 at Knotilus! L9a23 is $9^2_{22}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+2 a^5 z+2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-5 a^3 z-3 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -a z+a z^{-1} +z a^{-1}$ (db) Kauffman polynomial $-a^4 z^8-a^2 z^8-3 a^5 z^7-6 a^3 z^7-3 a z^7-2 a^6 z^6-4 a^4 z^6-6 a^2 z^6-4 z^6-a^7 z^5+8 a^5 z^5+13 a^3 z^5+a z^5-3 z^5 a^{-1} +4 a^6 z^4+13 a^4 z^4+15 a^2 z^4-z^4 a^{-2} +5 z^4+3 a^7 z^3-10 a^5 z^3-15 a^3 z^3+2 a z^3+4 z^3 a^{-1} -a^6 z^2-11 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -2 z^2-2 a^7 z+8 a^5 z+11 a^3 z-z a^{-1} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a z^{-1}$ (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 \
-6-5-4-3-2-10123χ
6         1-1
4        2 2
2       31 -2
0      42  2
-2     54   -1
-4    33    0
-6   35     2
-8  23      -1
-10  3       3
-1212        -1
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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