L11n292

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 Planar diagram presentation X6172 X11,18,12,19 X3849 X2,16,3,15 X16,7,17,8 X9,11,10,22 X17,1,18,4 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20 Gauss code {1, -4, -3, 7}, {-9, -1, 5, 3, -6, 8}, {-2, 9, -11, 10, 4, -5, -7, 2, -8, 11, -10, 6}

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-t(2)^{2}t(3)^{3}+2t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-2t(2)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)+t(2)^{2}-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}}$ (db) Jones polynomial ${\displaystyle q^{5}-3q^{4}-q^{-4}+5q^{3}+3q^{-3}-6q^{2}-4q^{-2}+8q+7q^{-1}-6}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{2}a^{-4}+a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-2a^{2}z^{2}-5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{6}+4z^{4}+5z^{2}-2z^{-2}+1}$ (db) Kauffman polynomial ${\displaystyle 2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}-2z^{7}a^{-1}+4z^{7}a^{-3}-14a^{2}z^{6}-21z^{6}a^{-2}+z^{6}a^{-4}-36z^{6}-4a^{3}z^{5}-6az^{5}-16z^{5}a^{-1}-14z^{5}a^{-3}+19a^{2}z^{4}+30z^{4}a^{-2}+z^{4}a^{-4}+48z^{4}+4a^{3}z^{3}+14az^{3}+24z^{3}a^{-1}+17z^{3}a^{-3}+3z^{3}a^{-5}-9a^{2}z^{2}-20z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}}$ (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 \
-5-4-3-2-101234χ
11         11
9        2 -2
7       31 2
5      43  -1
3     431  2
1    46    2
-1   331    1
-3  25      3
-5 12       -1
-7 2        2
-91         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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