L11a511

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 Planar diagram presentation X8192 X14,3,15,4 X20,12,21,11 X18,10,19,9 X22,16,13,15 X12,20,7,19 X10,22,11,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17 Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 4, -7, 3, -6}, {10, -2, 5, -8, 11, -4, 6, -3, 7, -5}
A Braid Representative

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-3t(1)^{2}t(2)t(3)^{2}+5t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(3)-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)+3t(1)^{2}t(2)t(3)-5t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{8}+3q^{7}-6q^{6}+11q^{5}-14q^{4}+17q^{3}-16q^{2}+15q-10+7q^{-1}-3q^{-2}+q^{-3}}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle -z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-5a^{-2}-2z^{4}-3z^{2}+z^{-2}+1}$ (db) Kauffman polynomial ${\displaystyle z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+5z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+6z^{8}a^{-6}-14z^{6}a^{-6}+17z^{4}a^{-6}-9z^{2}a^{-6}+2a^{-6}+4z^{9}a^{-5}-4z^{7}a^{-5}-2z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+z^{10}a^{-4}+10z^{8}a^{-4}-35z^{6}a^{-4}+53z^{4}a^{-4}-35z^{2}a^{-4}-a^{-4}z^{-2}+10a^{-4}+7z^{9}a^{-3}-13z^{7}a^{-3}+7z^{5}a^{-3}+5z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}+a^{2}z^{6}-27z^{6}a^{-2}-3a^{2}z^{4}+35z^{4}a^{-2}+3a^{2}z^{2}-29z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+12a^{-2}+3z^{9}a^{-1}+3az^{7}-z^{7}a^{-1}-8az^{5}-10z^{5}a^{-1}+5az^{3}+9z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+4z^{8}-8z^{6}+2z^{4}-2z^{2}-z^{-2}+4}$ (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          2 2
13         41 -3
11        72  5
9       85   -3
7      96    3
5     89     1
3    78      -1
1   49       5
-1  36        -3
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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