# L11n181

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n181 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-u^2 v^2+u^2 v+u^2-u v^3+3 u v^2-u v+v^4+v^3-v^2}{u v^2}$ (db) Jones polynomial $-q^{7/2}+2 q^{5/2}-\frac{2}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}+3 \sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z+2 a^5 z^{-1} -a^3 z^3-5 a^3 z-3 a^3 z^{-1} -z a^{-3} +a z^3+z^3 a^{-1} +a z+a z^{-1} +z a^{-1}$ (db) Kauffman polynomial $a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} -2 a^4 z^4+6 a^4 z^2-3 a^4-2 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-3 z^3 a^{-3} -9 a^3 z+z a^{-3} +3 a^3 z^{-1} +2 z^6 a^{-2} -7 z^4 a^{-2} +4 a^2 z^2+5 z^2 a^{-2} -3 a^2+z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} -3 a z^3-z^3 a^{-1} +z a^{-1} +a z^{-1} +2 z^6-5 z^4+3 z^2-1$ (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 \
-6-5-4-3-2-101234χ
8          11
6         1 -1
4        11 0
2       21  -1
0     121   0
-2     23    1
-4   121     0
-6    2      2
-8  11       0
-101          1
-121          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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