# L9n25

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n25 at Knotilus! L9n25 is $9^3_{18}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $- q^{-5} +2 q^{-4} -2 q^{-3} +q^2+3 q^{-2} -q-2 q^{-1} +4$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2-2 z^2-2 z^{-2} -3+ a^{-2} z^{-2} + a^{-2}$ (db) Kauffman polynomial $a^3 z^7+a z^7+2 a^4 z^6+3 a^2 z^6+z^6+a^5 z^5-2 a^3 z^5-3 a z^5-7 a^4 z^4-11 a^2 z^4-4 z^4-3 a^5 z^3-2 a^3 z^3+2 a z^3+z^3 a^{-1} +5 a^4 z^2+13 a^2 z^2+z^2 a^{-2} +9 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-1012χ
5       11
3        0
1     41 3
-1    24  2
-3   111  1
-5  12    1
-7 11     0
-9 1      1
-111       -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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