# L10n52

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n52 at Knotilus!

 Planar diagram presentation X8192 X9,19,10,18 X14,6,15,5 X16,12,17,11 X3,10,4,11 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X17,2,18,3 Gauss code {1, 10, -5, -9, 3, -8}, {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {-u^{2}v^{2}+2u^{2}v-u^{2}+uv^{4}-3uv^{3}+5uv^{2}-3uv+u-v^{4}+2v^{3}-v^{2}}{uv^{2}}}}$ (db) Jones polynomial ${\displaystyle -2q^{3/2}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}+3z^{3}a+5za+2az^{-1}-2za^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2a^{4}z^{8}-2a^{2}z^{8}-4a^{5}z^{7}-7a^{3}z^{7}-3az^{7}-3a^{6}z^{6}+2a^{4}z^{6}+4a^{2}z^{6}-z^{6}-a^{7}z^{5}+11a^{5}z^{5}+21a^{3}z^{5}+9az^{5}+7a^{6}z^{4}+2a^{4}z^{4}-7a^{2}z^{4}-2z^{4}+2a^{7}z^{3}-9a^{5}z^{3}-26a^{3}z^{3}-18az^{3}-3z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+4a^{2}z^{2}+3z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}}$ (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 \
-6-5-4-3-2-1012χ
4        22
2       2 -2
0      42 2
-2     43  -1
-4    43   1
-6   35    2
-8  23     -1
-10 13      2
-12 2       -2
-141        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

### Computer Talk

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