# L10n83

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n83 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $1-2 q^{-1} +5 q^{-2} -4 q^{-3} +6 q^{-4} -5 q^{-5} +5 q^{-6} -3 q^{-7} + q^{-8}$ (db) Signature -4 (db) HOMFLY-PT polynomial $z^4 a^6+2 z^2 a^6+a^6 z^{-2} +a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2$ (db) Kauffman polynomial $z^2 a^{10}-a^{10}+3 z^3 a^9-z a^9+2 z^6 a^8-5 z^4 a^8+9 z^2 a^8-3 a^8+3 z^7 a^7-9 z^5 a^7+10 z^3 a^7-3 z a^7+z^8 a^6+3 z^6 a^6-18 z^4 a^6+20 z^2 a^6+a^6 z^{-2} -7 a^6+5 z^7 a^5-15 z^5 a^5+9 z^3 a^5+z a^5-2 a^5 z^{-1} +z^8 a^4+2 z^6 a^4-17 z^4 a^4+18 z^2 a^4+2 a^4 z^{-2} -8 a^4+2 z^7 a^3-6 z^5 a^3+2 z^3 a^3+3 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 a^2$ (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 \
-6-5-4-3-2-1012χ
1        11
-1       1 -1
-3      41 3
-5     23  1
-7    42   2
-9  122    1
-11  44     0
-13 13      2
-15 2       -2
-171        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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