# L11n247

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n247 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) 0 (db) Jones polynomial $q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-a^3 z^3+z^3 a^{-3} -2 a^3 z+2 z a^{-3} +a z^5-z^5 a^{-1} +5 a z^3-5 z^3 a^{-1} +6 a z-6 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^4 z^6+z^6 a^{-4} -4 a^4 z^4-4 z^4 a^{-4} +2 a^4 z^2+2 z^2 a^{-4} +2 a^3 z^7+2 z^7 a^{-3} -10 a^3 z^5-10 z^5 a^{-3} +11 a^3 z^3+11 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^2 z^8+z^8 a^{-2} -5 a^2 z^6-5 z^6 a^{-2} +4 a^2 z^4+4 z^4 a^{-2} +3 a z^7+3 z^7 a^{-1} -19 a z^5-19 z^5 a^{-1} +29 a z^3+29 z^3 a^{-1} -12 a z-12 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8-12 z^6+16 z^4-4 z^2-1$ (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 \
-5-4-3-2-1012345χ
10          1-1
8         1 1
6       111 1
4      121  0
2     221   1
0    242    0
-2   122     1
-4  121      0
-6 111       1
-8 1         1
-101          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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