# L9a33

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a33 at Knotilus! L9a33 is $9^2_{24}$ in the Rolfsen table of links.  Alternate symmetric version, with three lines touching at center  Alternate symmetric version, with three lines touching at circumference  Form made from 45-degree lines and circular arcs.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^2-3 u^2 v+3 u^2-3 u v^2+7 u v-3 u+3 v^2-3 v+1}{u v}$ (db) Jones polynomial $-\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{10}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3+3 a^3 z-a z^5-2 a z^3+z^3 a^{-1} -3 a z$ (db) Kauffman polynomial $-a^4 z^8-a^2 z^8-3 a^5 z^7-7 a^3 z^7-4 a z^7-2 a^6 z^6-7 a^4 z^6-11 a^2 z^6-6 z^6-a^7 z^5+5 a^5 z^5+9 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^6 z^4+18 a^4 z^4+24 a^2 z^4-z^4 a^{-2} +8 z^4+3 a^7 z^3-3 a^5 z^3-2 a^3 z^3+8 a z^3+4 z^3 a^{-1} -11 a^4 z^2-14 a^2 z^2-3 z^2-3 a^7 z+a^5 z+2 a^3 z-2 a z-a^6+a^7 z^{-1} +a^5 z^{-1}$ (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        3 3
2       31 -2
0      53  2
-2     64   -2
-4    34    -1
-6   46     2
-8  23      -1
-10  4       4
-1212        -1
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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