# L9a41

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a41 at Knotilus! L9a41 is $9^2_{23}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1) t(2)+1) (t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^5 z^5+4 a^5 z^3+5 a^5 z+a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-11 a^3 z-a^3 z^{-1} +a z^5+4 a z^3+4 a z$ (db) Kauffman polynomial $-z^3 a^9+z a^9-2 z^4 a^8+z^2 a^8-3 z^5 a^7+2 z^3 a^7-z a^7-4 z^6 a^6+7 z^4 a^6-6 z^2 a^6-3 z^7 a^5+5 z^5 a^5-4 z^3 a^5+4 z a^5-a^5 z^{-1} -z^8 a^4-3 z^6 a^4+13 z^4 a^4-10 z^2 a^4+a^4-5 z^7 a^3+16 z^5 a^3-17 z^3 a^3+11 z a^3-a^3 z^{-1} -z^8 a^2+8 z^4 a^2-7 z^2 a^2-2 z^7 a+8 z^5 a-10 z^3 a+5 z a-z^6+4 z^4-4 z^2$ (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
4         1-1
2        1 1
0       21 -1
-2      41  3
-4     23   1
-6    43    1
-8   22     0
-10  24      -2
-12 13       2
-14 1        -1
-161         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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