L11a385

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 Planar diagram presentation X6172 X10,3,11,4 X16,12,17,11 X18,22,19,21 X20,14,21,13 X12,20,13,19 X22,18,9,17 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10 Gauss code {1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, 3, -6, 5, -9, 8, -3, 7, -4, 6, -5, 4, -7}
A Braid Representative

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+2t(3)^{3}+6t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+7t(2)t(3)^{2}-4t(3)^{2}-7t(1)t(3)+4t(1)t(2)t(3)-6t(2)t(3)+4t(3)+4t(1)-2t(1)t(2)+2t(2)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{8}+4q^{7}-10q^{6}+14q^{5}-18q^{4}+21q^{3}+q^{-3}-19q^{2}-2q^{-2}+17q+7q^{-1}-10}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-3a^{-2}+6a^{-4}-2a^{-6}-3+a^{2}z^{-2}-2a^{-2}z^{-2}+3a^{-4}z^{-2}-a^{-6}z^{-2}-z^{-2}}$ (db) Kauffman polynomial ${\displaystyle z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+8z^{8}a^{-2}+15z^{8}a^{-4}+10z^{8}a^{-6}+3z^{8}+2az^{7}+6z^{7}a^{-1}+z^{7}a^{-3}+6z^{7}a^{-5}+9z^{7}a^{-7}+a^{2}z^{6}-15z^{6}a^{-2}-31z^{6}a^{-4}-17z^{6}a^{-6}+4z^{6}a^{-8}-4z^{6}-4az^{5}-23z^{5}a^{-1}-33z^{5}a^{-3}-31z^{5}a^{-5}-16z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+6z^{4}a^{-2}+18z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}-2z^{4}+23z^{3}a^{-1}+50z^{3}a^{-3}+39z^{3}a^{-5}+11z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}-3z^{2}a^{-2}-4z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}+4az-10za^{-1}-34za^{-3}-27za^{-5}-7za^{-7}-4a^{2}+4a^{-2}+5a^{-4}+a^{-6}-3-2az^{-1}+2a^{-1}z^{-1}+10a^{-3}z^{-1}+8a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{2}z^{-2}-2a^{-2}z^{-2}-3a^{-4}z^{-2}-a^{-6}z^{-2}+z^{-2}}$ (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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        73  4
9       117   -4
7      107    3
5     911     2
3    810      -2
1   512       7
-1  25        -3
-3  5         5
-512          -1
-71           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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