# L9n14

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n14 at Knotilus! L9n14 is $9^2_{50}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^2-u^2 v+u v-v+1}{u v}$ (db) Jones polynomial $q^{9/2}-q^{7/2}+q^{5/2}-\frac{1}{q^{5/2}}-2 q^{3/2}+\frac{1}{q^{3/2}}+\sqrt{q}-\frac{2}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $z^3 a^{-3} +3 z a^{-3} + a^{-3} z^{-1} -z^5 a^{-1} +a z^3-5 z^3 a^{-1} +3 a z-7 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^7 a^{-1} -z^7 a^{-3} -2 z^6 a^{-2} -z^6 a^{-4} -z^6+5 z^5 a^{-1} +5 z^5 a^{-3} +9 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-2 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -a^2 z^2-10 z^2 a^{-2} -6 z^2 a^{-4} -5 z^2-a^3 z+4 a z+9 z a^{-1} +4 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-1}$ (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 \
-2-1012345χ
10       1-1
8        0
6     11 0
4    1   1
2    1   1
0  21    1
-2  1     1
-411      0
-61       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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