# L9n26

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n26 at Knotilus! L9n26 is $9^3_{19}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(2)-1) (t(3)-1) \left(t(3)^2-t(1)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^{-6} - q^{-5} +3 q^{-4} -2 q^{-3} +3 q^{-2} -q-2 q^{-1} +3$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^6-2 a^4 z^2-2 a^4 z^{-2} -4 a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +3 a^2-z^2$ (db) Kauffman polynomial $a^6 z^6-5 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -5 a^6+a^5 z^7-3 a^5 z^5-a^5 z^3+5 a^5 z-2 a^5 z^{-1} +4 a^4 z^6-17 a^4 z^4+20 a^4 z^2+2 a^4 z^{-2} -10 a^4+a^3 z^7-a^3 z^5-6 a^3 z^3+7 a^3 z-2 a^3 z^{-1} +3 a^2 z^6-12 a^2 z^4+15 a^2 z^2+a^2 z^{-2} -7 a^2+2 a z^5-5 a z^3+3 a z+z a^{-1} +3 z^2-1$ (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 \
-6-5-4-3-2-101χ
3       1-1
1      2 2
-1     23 1
-3    111 1
-5   12   1
-7  21    1
-9 13     2
-11        0
-131       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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