L11n455

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 Planar diagram presentation X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X7,19,8,18 X17,9,18,8 X10,13,5,14 X19,22,20,17 X21,11,22,16 X11,21,12,20 X4,16,1,15 Gauss code {1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, {-6, 5, -8, 10, -9, 8}

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}+t(1)t(4)t(2)^{2}-t(1)t(3)t(4)t(2)^{2}-2t(3)^{2}t(2)+t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)+t(3)^{2}t(4)t(2)-2t(1)t(4)t(2)+t(1)t(3)t(4)t(2)-t(3)t(4)t(2)+t(3)^{2}-t(3)-t(3)^{2}t(4)+t(1)t(4)+2t(3)t(4)-t(4)}{{\sqrt {t(1)}}t(2)t(3){\sqrt {t(4)}}}}}$ (db) Jones polynomial ${\displaystyle 2q^{9/2}-6q^{7/2}+{\frac {1}{q^{7/2}}}+5q^{5/2}-{\frac {4}{q^{5/2}}}-9q^{3/2}+{\frac {5}{q^{3/2}}}-q^{11/2}+7{\sqrt {q}}-{\frac {8}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle -2z^{5}a^{-1}+3az^{3}-8z^{3}a^{-1}+3z^{3}a^{-3}-a^{3}z+6az-12za^{-1}+8za^{-3}-za^{-5}+3az^{-1}-8a^{-1}z^{-1}+7a^{-3}z^{-1}-2a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}}$ (db) Kauffman polynomial ${\displaystyle -z^{9}a^{-1}-z^{9}a^{-3}-6z^{8}a^{-2}-2z^{8}a^{-4}-4z^{8}-5az^{7}-7z^{7}a^{-1}-3z^{7}a^{-3}-z^{7}a^{-5}-2a^{2}z^{6}+19z^{6}a^{-2}+7z^{6}a^{-4}+10z^{6}+17az^{5}+39z^{5}a^{-1}+27z^{5}a^{-3}+5z^{5}a^{-5}+2a^{2}z^{4}-6z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}-4a^{3}z^{3}-24az^{3}-52z^{3}a^{-1}-42z^{3}a^{-3}-10z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-18z^{2}a^{-2}-9z^{2}a^{-4}-10z^{2}+2a^{3}z+16az+31za^{-1}+27za^{-3}+10za^{-5}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}}$ (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 \
-3-2-10123456χ
12         11
10        1 -1
8       51 4
6      23  1
4     73   4
2   134    2
0   65     1
-2 124      3
-4 34       -1
-6 3        3
-81         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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