# L9a36

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a36 at Knotilus! L9a36 is $9^2_{1}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^3-u^3 v^2-u^2 v^3+u^2 v^2-u^2 v-u v^2+u v-u-v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+2 q^{7/2}-2 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+2 q^{15/2}-3 q^{13/2}+3 q^{11/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $z^5 a^{-7} +4 z^3 a^{-7} +3 z a^{-7} -z^7 a^{-5} -6 z^5 a^{-5} -11 z^3 a^{-5} -7 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1}$ (db) Kauffman polynomial $-z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} +5 z^6 a^{-4} +3 z^6 a^{-6} -2 z^6 a^{-8} +6 z^5 a^{-3} +15 z^5 a^{-5} +7 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-4} +4 z^4 a^{-8} -2 z^4 a^{-10} -11 z^3 a^{-3} -21 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -2 z^3 a^{-11} -2 z^2 a^{-6} +z^2 a^{-10} -z^2 a^{-12} +7 z a^{-3} +9 z a^{-5} +2 z a^{-7} +z a^{-9} +z a^{-11} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 \
-2-101234567χ
20         1-1
18        1 1
16       11 0
14      21  1
12     22   0
10    11    0
8   12     1
6  11      0
4 12       1
2          0
01         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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