# L11n208

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n208 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^3-u^3 v^2+u^2 v^2+u v-v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $\frac{1}{q^{9/2}}+q^{7/2}-\frac{1}{q^{7/2}}-q^{5/2}+\frac{1}{q^{5/2}}+q^{3/2}-\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}-2 \sqrt{q}+\frac{1}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^3 z^5+5 a^3 z^3+6 a^3 z+2 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -16 a z^3+5 z^3 a^{-1} -14 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8+7 a z^7+7 z^7 a^{-1} +6 a^2 z^6+7 z^6 a^{-2} +13 z^6-2 a^3 z^5-18 a z^5-16 z^5 a^{-1} -a^4 z^4-12 a^2 z^4-15 z^4 a^{-2} -26 z^4-a^5 z^3+8 a^3 z^3+24 a z^3+15 z^3 a^{-1} -a^6 z^2+2 a^4 z^2+11 a^2 z^2+10 z^2 a^{-2} +18 z^2-a^7 z+a^5 z-7 a^3 z-16 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (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 \
-4-3-2-1012345χ
8         1-1
6          0
4       11 0
2      1   1
0      1   1
-2    21    1
-4    1     1
-6  11      0
-8          0
-1011        0
-121         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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