# L10n25

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n25 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3-u v^2-2 u v+u+v^5-2 v^4-v^3+v^2}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-q^{7/2}-\frac{2}{q^{7/2}}+q^{5/2}+\frac{1}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^5+2 a^5 z^{-1} -2 z^3 a^3-7 z a^3-4 a^3 z^{-1} +z^5 a+5 z^3 a+6 z a+3 a z^{-1} -z a^{-1} - a^{-1} z^{-1} -z a^{-3}$ (db) Kauffman polynomial $-a^4 z^8-a^2 z^8-a^5 z^7-3 a^3 z^7-2 a z^7+5 a^4 z^6+5 a^2 z^6-z^6 a^{-2} -z^6+6 a^5 z^5+18 a^3 z^5+13 a z^5-z^5 a^{-3} -5 a^4 z^4-3 a^2 z^4+5 z^4 a^{-2} +7 z^4-11 a^5 z^3-30 a^3 z^3-22 a z^3+z^3 a^{-1} +4 z^3 a^{-3} -a^4 z^2-5 a^2 z^2-5 z^2 a^{-2} -9 z^2+8 a^5 z+18 a^3 z+13 a z+z a^{-1} -2 z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-101234χ
8          11
6           0
4       111 -1
2      11   0
0     121   0
-2    122    1
-4   11      0
-6  111      1
-8 12        1
-10           0
-121          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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