# L9a24

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a24 at Knotilus! L9a24 is $9^2_{21}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^3-u^2 v^2+u^2 v-u^2+u v^4-3 u v^3+3 u v^2-3 u v+u-v^4+v^3-v^2+v}{u v^2}$ (db) Jones polynomial $-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+3 z^3 a^5+4 z a^5+3 a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-2 a^3 z^{-1} -z^3 a-2 z a$ (db) Kauffman polynomial $a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-8 a^6 z^4+11 a^6 z^2-3 a^6+5 a^5 z^7-15 a^5 z^5+22 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8+a^4 z^6-6 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-4 a^3 z^5+3 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +2 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-3 a z^3+2 a z$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        1 -1
-2       31 2
-4      32  -1
-6     42   2
-8    34    1
-10   33     0
-12  13      2
-14 13       -2
-16 1        1
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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