# L11n192

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n192 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u v-u-v+2) (2 u v-u-v+1)}{u v}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-9 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-a^3 z^3-z^3 a^{-3} -a^3 z-z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +2 a z^3+2 z^3 a^{-1} +a z+2 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-5} -z a^{-5} +a^4 z^6-3 a^4 z^4+3 z^4 a^{-4} +2 a^4 z^2-2 z^2 a^{-4} +3 a^3 z^7+z^7 a^{-3} -10 a^3 z^5+8 a^3 z^3+2 z^3 a^{-3} -2 a^3 z- a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -8 a^2 z^6-5 z^6 a^{-2} +3 a^2 z^4+9 z^4 a^{-2} -4 z^2 a^{-2} + a^{-2} +a z^9+z^9 a^{-1} +3 a z^7+z^7 a^{-1} -16 a z^5-6 z^5 a^{-1} +13 a z^3+6 z^3 a^{-1} -3 a z- a^{-1} z^{-1} +5 z^8-14 z^6+12 z^4-4 z^2$ (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 \
-5-4-3-2-101234χ
10         11
8        2 -2
6       41 3
4      43  -1
2     53   2
0    45    1
-2   34     -1
-4  24      2
-6 13       -2
-8 2        2
-101         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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