# L9n3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n3 at Knotilus! L9n3 is $9^2_{47}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z-a z-a z^{-1}$ (db) Kauffman polynomial $a^8 z^6-5 a^8 z^4+6 a^8 z^2-2 a^8+a^7 z^7-5 a^7 z^5+6 a^7 z^3-2 a^7 z+a^7 z^{-1} +2 a^6 z^6-10 a^6 z^4+13 a^6 z^2-5 a^6+a^5 z^7-5 a^5 z^5+7 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +a^4 z^6-5 a^4 z^4+7 a^4 z^2-3 a^4+a^3 z^3-2 a^3 z+a^2+a z-a z^{-1}$ (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 \
-7-6-5-4-3-2-10χ
0       11
-2      121
-4     1 12
-6    12  1
-8   11   0
-10   11   0
-12 11     0
-14        0
-161       -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$

### Computer Talk

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