# L9n18

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n18 at Knotilus! L9n18 is $9^2_{53}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{\left(t(2) t(1)^2+1\right) \left(t(1) t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{21/2}}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{11} (-z)+a^9 z^5+6 a^9 z^3+8 a^9 z+a^9 z^{-1} -a^7 z^7-7 a^7 z^5-15 a^7 z^3-11 a^7 z-a^7 z^{-1}$ (db) Kauffman polynomial $-z^3 a^{13}+3 z a^{13}+z a^{11}-z^6 a^{10}+6 z^4 a^{10}-8 z^2 a^{10}-z^7 a^9+7 z^5 a^9-14 z^3 a^9+9 z a^9-a^9 z^{-1} -z^6 a^8+6 z^4 a^8-8 z^2 a^8+a^8-z^7 a^7+7 z^5 a^7-15 z^3 a^7+11 z a^7-a^7 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-6        11
-8        11
-10      1  1
-12    1    1
-14   111   -1
-16   1     -1
-18  11     0
-201        1
-221        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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