# L10n56

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n56 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (u+1)^2 (v-1)}{u^{3/2} \sqrt{v}}$ (db) Jones polynomial $-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^5 (-z)-a^3 z+a z^5+5 a z^3-z^3 a^{-1} +5 a z+a z^{-1} -3 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^2 z^8-z^8-a^3 z^7-2 a z^7-z^7 a^{-1} +6 a^2 z^6+6 z^6-a^5 z^5+6 a^3 z^5+13 a z^5+6 z^5 a^{-1} -a^6 z^4-8 a^2 z^4-9 z^4+4 a^5 z^3-8 a^3 z^3-22 a z^3-10 z^3 a^{-1} +3 a^6 z^2+a^4 z^2+a^2 z^2+3 z^2-2 a^5 z+2 a^3 z+10 a z+6 z a^{-1} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-101234χ
6         11
4          0
2       11 0
0     21   1
-2    121   0
-4   111    1
-6   1      1
-8 111      1
-10          0
-121         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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