# L10n36

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n36 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $0$ (db) Jones polynomial $q^{9/2}+\frac{1}{q^{9/2}}-q^{7/2}-\frac{1}{q^{7/2}}+q^{5/2}+\frac{1}{q^{5/2}}-q^{3/2}-\frac{1}{q^{3/2}}-\sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-a^3 z^3+z^3 a^{-3} -3 a^3 z+3 z a^{-3} -2 a^3 z^{-1} +2 a^{-3} z^{-1} +a z^5-z^5 a^{-1} +6 a z^3-6 z^3 a^{-1} +11 a z-11 z a^{-1} +7 a z^{-1} -7 a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^7-a z^7-z^7 a^{-1} -z^7 a^{-3} -a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} -z^6 a^{-4} -2 z^6+5 a^3 z^5+7 a z^5+7 z^5 a^{-1} +5 z^5 a^{-3} +5 a^4 z^4+12 a^2 z^4+12 z^4 a^{-2} +5 z^4 a^{-4} +14 z^4-6 a^3 z^3-14 a z^3-14 z^3 a^{-1} -6 z^3 a^{-3} -6 a^4 z^2-18 a^2 z^2-18 z^2 a^{-2} -6 z^2 a^{-4} -24 z^2+4 a^3 z+14 a z+14 z a^{-1} +4 z a^{-3} +2 a^4+8 a^2+8 a^{-2} +2 a^{-4} +13-2 a^3 z^{-1} -7 a z^{-1} -7 a^{-1} z^{-1} -2 a^{-3} z^{-1}$ (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 \
-5-4-3-2-1012345χ
10          1-1
8           0
6        11 0
4      11   0
2     1 1   2
0    141    2
-2   1 1     2
-4   11      0
-6 11        0
-8           0
-101          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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