L11a429

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 Planar diagram presentation X6172 X12,4,13,3 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X22,20,11,19 X8,22,9,21 X20,8,21,7 X2,9,3,10 X4,18,1,17 Gauss code {1, -10, 2, -11}, {5, -1, 9, -8, 10, -6}, {3, -2, 4, -5, 6, -4, 11, -3, 7, -9, 8, -7}
A Braid Representative
A Morse Link Presentation

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-5t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-4t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+9t(1)t(3)t(2)-9t(3)t(2)+4t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-4t(1)t(3)+5t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}}$ (db) Jones polynomial ${\displaystyle q^{9}-4q^{8}+9q^{7}-14q^{6}+20q^{5}-22q^{4}+23q^{3}-19q^{2}-q^{-2}+15q+4q^{-1}-8}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+4z^{2}a^{-2}-4z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-2}-6a^{-4}+2a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}}$ (db) Kauffman polynomial ${\displaystyle z^{6}a^{-10}-2z^{4}a^{-10}+z^{2}a^{-10}+4z^{7}a^{-9}-9z^{5}a^{-9}+5z^{3}a^{-9}+7z^{8}a^{-8}-16z^{6}a^{-8}+11z^{4}a^{-8}-4z^{2}a^{-8}+a^{-8}+6z^{9}a^{-7}-7z^{7}a^{-7}-4z^{5}a^{-7}+3z^{3}a^{-7}+2z^{10}a^{-6}+12z^{8}a^{-6}-34z^{6}a^{-6}+23z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+12z^{9}a^{-5}-18z^{7}a^{-5}+6z^{5}a^{-5}-5z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+13z^{8}a^{-4}-28z^{6}a^{-4}+12z^{4}a^{-4}+8z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+6z^{9}a^{-3}-9z^{5}a^{-3}+z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+8z^{8}a^{-2}-7z^{6}a^{-2}-4z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}+7z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-6z^{4}+3z^{2}}$ (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 \
-3-2-1012345678χ
19           11
17          3 -3
15         61 5
13        94  -5
11       115   6
9      119    -2
7     1211     1
5    812      4
3   711       -4
1  310        7
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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