# L10n4

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n4 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^5-2 u v^4+2 u v^3-2 u v^2-2 v^3+2 v^2-2 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-q^{7/2}+2 q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-a^3 z^5+6 a z^5-z^5 a^{-1} -5 a^3 z^3+12 a z^3-4 z^3 a^{-1} +a^5 z-8 a^3 z+9 a z-4 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $3 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6-2 a^4 z^4+3 a^4 z^2-2 a^4+2 a^3 z^7-9 a^3 z^5+z^5 a^{-3} +20 a^3 z^3-3 z^3 a^{-3} -16 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^8-2 a^2 z^6+2 z^6 a^{-2} -6 z^4 a^{-2} +7 a^2 z^2+3 z^2 a^{-2} -3 a^2- a^{-2} +4 a z^7+2 z^7 a^{-1} -16 a z^5-6 z^5 a^{-1} +25 a z^3+5 z^3 a^{-1} -13 a z-3 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +z^8-z^6-4 z^4+7 z^2-3$ (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
8        11
6       1 -1
4      21 1
2     21  -1
0    32   1
-2   33    0
-4  12     -1
-6 13      2
-811       0
-102        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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