# L11n243

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n243 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^3 v^2-4 u^3 v+u^3-2 u^2 v^2+2 u^2 v+2 u v^2-2 u v+v^3-4 v^2+2 v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{7/2}-3 q^{5/2}+5 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+2 a^5 z^{-1} -3 a^3 z^3-8 a^3 z-5 a^3 z^{-1} +z a^{-3} + a^{-3} z^{-1} +2 a z^5+8 a z^3-3 z^3 a^{-1} +11 a z+6 a z^{-1} -7 z a^{-1} -4 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^5 z^7-5 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+8 a^4 z^4-a^4 z^2+z^2 a^{-4} -a^4- a^{-4} +a^3 z^9+2 a^3 z^7-23 a^3 z^5+36 a^3 z^3+3 z^3 a^{-3} -22 a^3 z-3 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +6 a^2 z^8-22 a^2 z^6+2 z^6 a^{-2} +18 a^2 z^4-5 z^4 a^{-2} -2 a^2 z^2+8 z^2 a^{-2} -a^2-3 a^{-2} +a z^9+6 a z^7+5 z^7 a^{-1} -38 a z^5-20 z^5 a^{-1} +54 a z^3+30 z^3 a^{-1} -30 a z-18 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +4 z^8-12 z^6+5 z^4+6 z^2-3$ (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
8         1-1
6        2 2
4       31 -2
2      42  2
0     44   0
-2    341   0
-4   34     1
-6  23      -1
-8 14       3
-10 1        -1
-121         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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