# L10n113

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n113 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u v w+u v x y-u v y+u v+u w y-u x y+v w-v x-w x y+w x-w+x y}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x} \sqrt{y}}$ (db) Jones polynomial $q^6-q^5+5 q^4-q^3+ q^{-3} +5 q^2+ q^{-1} +5$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^{-6} z^{-4} +2 a^{-6} z^{-2} + a^{-6} -4 a^{-4} z^{-4} -3 z^2 a^{-4} -9 a^{-4} z^{-2} -8 a^{-4} +2 z^4 a^{-2} +a^2 z^{-4} +6 a^{-2} z^{-4} +a^2 z^2+9 z^2 a^{-2} +3 a^2 z^{-2} +15 a^{-2} z^{-2} +3 a^2+16 a^{-2} -z^4-4 z^{-4} -7 z^2-11 z^{-2} -12$ (db) Kauffman polynomial $z^6 a^{-6} -5 z^4 a^{-6} - a^{-6} z^{-4} +10 z^2 a^{-6} +5 a^{-6} z^{-2} -10 a^{-6} +z^7 a^{-5} -z^5 a^{-5} -10 z^3 a^{-5} +4 a^{-5} z^{-3} +20 z a^{-5} -15 a^{-5} z^{-1} +6 z^6 a^{-4} -25 z^4 a^{-4} -4 a^{-4} z^{-4} +30 z^2 a^{-4} +14 a^{-4} z^{-2} -25 a^{-4} +z^7 a^{-3} +3 z^5 a^{-3} -30 z^3 a^{-3} +12 a^{-3} z^{-3} +55 z a^{-3} -41 a^{-3} z^{-1} +a^2 z^6+5 z^6 a^{-2} -6 a^2 z^4-25 z^4 a^{-2} -a^2 z^{-4} -6 a^{-2} z^{-4} +10 a^2 z^2+40 z^2 a^{-2} +5 a^2 z^{-2} +18 a^{-2} z^{-2} -10 a^2-31 a^{-2} +a z^5+5 z^5 a^{-1} -10 a z^3-30 z^3 a^{-1} +4 a z^{-3} +12 a^{-1} z^{-3} +20 a z+55 z a^{-1} -15 a z^{-1} -41 a^{-1} z^{-1} +z^6-11 z^4-4 z^{-4} +30 z^2+14 z^{-2} -25$ (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 \
-4-3-2-10123456χ
13          11
11           0
9        51 4
7       15  4
5      4    4
3    4 1    5
1   1104     5
-1    6      6
-3  1        1
-51          1
-71          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{6}$ ${\mathbb Z}^{10}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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