# L10n3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n3 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $\frac{1}{q^{9/2}}+q^{7/2}-\frac{2}{q^{7/2}}-2 q^{5/2}+\frac{3}{q^{5/2}}+3 q^{3/2}-\frac{4}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-a^3 z^3-2 a^3 z+z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +a z^5+4 a z^3-2 z^3 a^{-1} +6 a z-5 z a^{-1} +4 a z^{-1} -4 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^4 z^6-4 a^4 z^4+4 a^4 z^2+z^2 a^{-4} -a^4- a^{-4} +2 a^3 z^7-8 a^3 z^5+8 a^3 z^3+2 z^3 a^{-3} -4 a^3 z-2 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^8-a^2 z^6+z^6 a^{-2} -8 a^2 z^4-3 z^4 a^{-2} +11 a^2 z^2+7 z^2 a^{-2} -4 a^2-4 a^{-2} +4 a z^7+2 z^7 a^{-1} -16 a z^5-8 z^5 a^{-1} +19 a z^3+13 z^3 a^{-1} -11 a z-9 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +z^8-z^6-7 z^4+13 z^2-7$ (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 \
-5-4-3-2-10123χ
8        1-1
6       1 1
4      21 -1
2     31  2
0    24   2
-2   221   1
-4  12     1
-6 12      -1
-8 1       1
-101        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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