# L10n96

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n96 at Knotilus!

 Planar diagram presentation X6172 X2536 X13,15,14,20 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X11,19,12,18 X19,13,20,12 X17,9,18,14 Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -8, 9, -3, 10}, {-7, 6, -10, 8, -9, 3}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-uvw^{2}x-uvwx^{2}+2uvwx-uvw-uvx+uv+uwx^{2}-uwx+ux+vw^{2}x-vwx+vw+w^{2}x^{2}-w^{2}x-wx^{2}+2wx-w-x}{{\sqrt {u}}{\sqrt {v}}wx}}}$ (db) Jones polynomial ${\displaystyle -q^{19/2}+2q^{17/2}-5q^{15/2}+5q^{13/2}-8q^{11/2}+6q^{9/2}-7q^{7/2}+3q^{5/2}-3q^{3/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle -2z^{5}a^{-5}+3z^{3}a^{-3}-9z^{3}a^{-5}+3z^{3}a^{-7}+8za^{-3}-16za^{-5}+9za^{-7}-za^{-9}+5a^{-3}z^{-1}-12a^{-5}z^{-1}+9a^{-7}z^{-1}-2a^{-9}z^{-1}+a^{-3}z^{-3}-3a^{-5}z^{-3}+3a^{-7}z^{-3}-a^{-9}z^{-3}}$ (db) Kauffman polynomial ${\displaystyle -z^{8}a^{-6}-z^{8}a^{-8}-4z^{7}a^{-5}-6z^{7}a^{-7}-2z^{7}a^{-9}-3z^{6}a^{-4}-4z^{6}a^{-6}-3z^{6}a^{-8}-2z^{6}a^{-10}+15z^{5}a^{-5}+18z^{5}a^{-7}+2z^{5}a^{-9}-z^{5}a^{-11}+9z^{4}a^{-4}+18z^{4}a^{-6}+13z^{4}a^{-8}+4z^{4}a^{-10}-6z^{3}a^{-3}-31z^{3}a^{-5}-26z^{3}a^{-7}+2z^{3}a^{-9}+3z^{3}a^{-11}-17z^{2}a^{-4}-33z^{2}a^{-6}-16z^{2}a^{-8}+13za^{-3}+28za^{-5}+21za^{-7}+3za^{-9}-3za^{-11}+13a^{-4}+24a^{-6}+11a^{-8}-a^{-10}-6a^{-3}z^{-1}-14a^{-5}z^{-1}-12a^{-7}z^{-1}-3a^{-9}z^{-1}+a^{-11}z^{-1}-3a^{-4}z^{-2}-6a^{-6}z^{-2}-3a^{-8}z^{-2}+a^{-3}z^{-3}+3a^{-5}z^{-3}+3a^{-7}z^{-3}+a^{-9}z^{-3}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
012345678χ
20        11
18       1 -1
16      41 3
14     44  0
12    41   3
10   35    2
8  43     1
6  4      4
433       0
23        3
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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