L11n194

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 Planar diagram presentation X8192 X16,6,17,5 X9,20,10,21 X21,10,22,11 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17 Gauss code {1, -7, 8, -9, 2, -11}, {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {3t(2)^{2}t(1)^{2}-4t(2)t(1)^{2}+t(1)^{2}-4t(2)^{2}t(1)+9t(2)t(1)-4t(1)+t(2)^{2}-4t(2)+3}{t(1)t(2)}}}$ (db) Jones polynomial ${\displaystyle 2q^{5/2}-5q^{3/2}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{5}z^{3}+a^{5}z-a^{3}z^{5}-a^{3}z^{3}+a^{3}z+a^{3}z^{-1}-2az^{5}-6az^{3}+2z^{3}a^{-1}-6az-az^{-1}+3za^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{3}z^{9}-az^{9}-3a^{4}z^{8}-5a^{2}z^{8}-2z^{8}-4a^{5}z^{7}-5a^{3}z^{7}-2az^{7}-z^{7}a^{-1}-3a^{6}z^{6}+a^{4}z^{6}+6a^{2}z^{6}+2z^{6}-a^{7}z^{5}+6a^{5}z^{5}+8a^{3}z^{5}-2az^{5}-3z^{5}a^{-1}+6a^{6}z^{4}+4a^{4}z^{4}-4a^{2}z^{4}-3z^{4}a^{-2}-5z^{4}+2a^{7}z^{3}-a^{5}z^{3}+11az^{3}+8z^{3}a^{-1}-3a^{6}z^{2}-a^{4}z^{2}+3a^{2}z^{2}+4z^{2}a^{-2}+5z^{2}-a^{7}z-3a^{3}z-9az-5za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}}$ (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 \
-6-5-4-3-2-10123χ
6         2-2
4        3 3
2       42 -2
0      73  4
-2     55   0
-4    66    0
-6   46     2
-8  25      -3
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

Computer Talk

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