# L11n343

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n343 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(v-1) (w-1) \left(u v-2 u+2 v^2-v\right)}{\sqrt{u} v^{3/2} \sqrt{w}}$ (db) Jones polynomial $-2 q^3+5 q^2-6 q+8-8 q^{-1} +8 q^{-2} -5 q^{-3} +4 q^{-4} - q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^6-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+a^2 z^{-2} +2 z^4+4 z^2+3-2 z^2 a^{-2} - a^{-2}$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8+a^5 z^7-2 a^3 z^7+3 z^7 a^{-1} +a^6 z^6-12 a^2 z^6+z^6 a^{-2} -10 z^6-2 a^5 z^5+4 a^3 z^5-6 z^5 a^{-1} -5 a^6 z^4-9 a^4 z^4+15 a^2 z^4+4 z^4 a^{-2} +23 z^4-3 a^5 z^3-10 a^3 z^3-a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} +8 a^6 z^2+13 a^4 z^2-8 a^2 z^2-7 z^2 a^{-2} -20 z^2+6 a^5 z+8 a^3 z-4 z a^{-1} -2 z a^{-3} -5 a^6-8 a^4-a^2+2 a^{-2} +5-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2}$ (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 \
-6-5-4-3-2-10123χ
7         2-2
5        3 3
3       32 -1
1      53  2
-1     55   0
-3    33    0
-5   25     3
-7  23      -1
-9 14       3
-11          0
-131         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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