# L11n100

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n100 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v^3-3 u v^2+u v-u-v^3+v^2-3 v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $\frac{3}{q^{9/2}}-\frac{4}{q^{7/2}}-q^{5/2}+\frac{4}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}-\frac{2}{q^{11/2}}-3 \sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 z^{-1} -a^5 z^3-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+2 a^3 z^{-1} +a z^5+3 a z^3-z^3 a^{-1} +a z-a z^{-1} -2 z a^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8+4 a^3 z^7+3 a z^7-z^7 a^{-1} +2 a^4 z^6+12 a^2 z^6+10 z^6-4 a^5 z^5-9 a^3 z^5+5 z^5 a^{-1} -a^6 z^4+a^4 z^4-12 a^2 z^4-14 z^4+11 a^5 z^3+17 a^3 z^3-a z^3-7 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+4 a^2 z^2+5 z^2-3 a^7 z-9 a^5 z-10 a^3 z-2 a z+2 z a^{-1} +a^4+a^7 z^{-1} +2 a^5 z^{-1} +2 a^3 z^{-1} +a z^{-1}$ (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 \
-4-3-2-101234χ
6        11
4       1 -1
2      21 1
0     21  -1
-2    32   1
-4   23    1
-6  22     0
-8 12      1
-1012       -1
-122        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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