# L10n99

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n99 at Knotilus!

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(x-1)\left(uvwx-uvw-2uvx-uwx+ux-vwx+vx+2wx+x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{7/2}+2q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {3}{q^{9/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{5}z^{-3}+2a^{5}z^{-1}+a^{3}z^{3}-3a^{3}z^{-3}-3a^{3}z-6a^{3}z^{-1}-za^{-3}-a^{-3}z^{-1}-az^{5}-az^{3}+3az^{-3}+2z^{3}a^{-1}-a^{-1}z^{-3}+2az+5az^{-1}+2za^{-1}}$ (db) Kauffman polynomial ${\displaystyle 6a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+6a^{5}z^{-1}+3a^{4}z^{6}-6a^{4}z^{4}+15a^{4}z^{2}+3a^{4}z^{-2}-13a^{4}+4a^{3}z^{7}-9a^{3}z^{5}+z^{5}a^{-3}+19a^{3}z^{3}-3z^{3}a^{-3}-3a^{3}z^{-3}-23a^{3}z+3za^{-3}+14a^{3}z^{-1}-a^{-3}z^{-1}+a^{2}z^{8}+8a^{2}z^{6}+2z^{6}a^{-2}-24a^{2}z^{4}-3z^{4}a^{-2}+33a^{2}z^{2}+6a^{2}z^{-2}-24a^{2}+a^{-2}+7az^{7}+3z^{7}a^{-1}-15az^{5}-5z^{5}a^{-1}+19az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-20az-4za^{-1}+12az^{-1}+3a^{-1}z^{-1}+z^{8}+7z^{6}-21z^{4}+18z^{2}+3z^{-2}-11}$ (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 \
-4-3-2-101234χ
8        11
6       21-1
4      4  4
2     32  -1
0    64   2
-2   57    2
-4  42     2
-6  5      5
-834       -1
-103        3
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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