# L10n57

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n57 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 $\frac{1}{q^{9/2}}-q^{7/2}-\frac{1}{q^{7/2}}+q^{5/2}-\frac{1}{q^{3/2}}-\frac{1}{\sqrt{q}}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-a^3 z^3-3 a^3 z-z a^{-3} +a z^5+5 a z^3+5 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^7-a z^7-a^4 z^6-a^2 z^6-z^6 a^{-2} -z^6+6 a^3 z^5+7 a z^5-z^5 a^{-3} +5 a^4 z^4+6 a^2 z^4+5 z^4 a^{-2} +6 z^4-10 a^3 z^3-14 a z^3+4 z^3 a^{-3} -5 a^4 z^2-7 a^2 z^2-5 z^2 a^{-2} -7 z^2+6 a^3 z+10 a z+2 z a^{-1} -2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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
8         11
6          0
4      111 -1
2     11   0
0    131   1
-2   112    2
-4   1      1
-6 111      1
-8          0
-101         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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