# L11n194

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n194 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2-4 t(2)^2 t(1)+9 t(2) t(1)-4 t(1)+t(2)^2-4 t(2)+3}{t(1) t(2)}$ (db) Jones polynomial $2 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z-a^3 z^5-a^3 z^3+a^3 z+a^3 z^{-1} -2 a z^5-6 a z^3+2 z^3 a^{-1} -6 a z-a z^{-1} +3 z a^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-3 a^4 z^8-5 a^2 z^8-2 z^8-4 a^5 z^7-5 a^3 z^7-2 a z^7-z^7 a^{-1} -3 a^6 z^6+a^4 z^6+6 a^2 z^6+2 z^6-a^7 z^5+6 a^5 z^5+8 a^3 z^5-2 a z^5-3 z^5 a^{-1} +6 a^6 z^4+4 a^4 z^4-4 a^2 z^4-3 z^4 a^{-2} -5 z^4+2 a^7 z^3-a^5 z^3+11 a z^3+8 z^3 a^{-1} -3 a^6 z^2-a^4 z^2+3 a^2 z^2+4 z^2 a^{-2} +5 z^2-a^7 z-3 a^3 z-9 a z-5 z a^{-1} -a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-6-5-4-3-2-10123χ
6         2-2
4        3 3
2       42 -2
0      73  4
-2     55   0
-4    66    0
-6   46     2
-8  25      -3
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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