# L10n51

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n51 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 u^2 v^2+u^2 v+u v^4-3 u v^3+3 u v^2-3 u v+u+v^3-2 v^2}{u v^2}$ (db) Jones polynomial $-\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -2 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -2 a^3 z^3-4 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $-z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-2 z^7 a^9+4 z^5 a^9-2 z^3 a^9+3 z a^9-a^9 z^{-1} -z^8 a^8+2 z^4 a^8-4 z^7 a^7+11 z^5 a^7-15 z^3 a^7+9 z a^7-2 a^7 z^{-1} -z^8 a^6+z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6-2 z^7 a^5+6 z^5 a^5-13 z^3 a^5+9 z a^5-2 a^5 z^{-1} -z^6 a^4-3 z^3 a^3+5 z a^3-a^3 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-2        22
-4       21-1
-6      31 2
-8     32  -1
-10    33   0
-12   23    1
-14  23     -1
-16 13      2
-18 1       -1
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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