# L9n12

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n12 at Knotilus! L9n12 is $9^2_{59}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^5-t(1) t(2)^3+t(2)^3+t(1) t(2)^2-t(2)^2-t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-5} +2 z a^{-3} +2 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-6 z^3 a^{-1} +4 a z-9 z a^{-1} +3 a z^{-1} -5 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +z^4 a^{-4} -2 z^2 a^{-4} +z^3 a^{-3} -3 z a^{-3} +2 a^{-3} z^{-1} +z^6 a^{-2} -7 z^4 a^{-2} +13 z^2 a^{-2} -5 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-8 z^5 a^{-1} +15 a z^3+19 z^3 a^{-1} -12 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +z^6-7 z^4+12 z^2-5$ (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 \
-4-3-2-1012345χ
12         1-1
10          0
8       11 0
6     11   0
4     11   0
2   121    0
0    2     2
-2  1       1
-41         1
-61         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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