# L11n327

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {uv^{2}w^{3}-2uv^{2}w^{2}-uvw^{3}+3uvw^{2}-2uvw-uw^{2}+uw+v^{3}\left(-w^{2}\right)+v^{3}w+2v^{2}w^{2}-3v^{2}w+v^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}}$ (db) Jones polynomial ${\displaystyle 2q^{5}-4q^{4}+6q^{3}-6q^{2}+8q-6+6q^{-1}-3q^{-2}+2q^{-3}-q^{-4}}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle 2z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-a^{2}z^{4}-3z^{4}a^{-2}-3a^{2}z^{2}-10z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+z^{6}+5z^{4}+10z^{2}+z^{-2}+8}$ (db) Kauffman polynomial ${\displaystyle 3z^{2}a^{-6}-a^{-6}+z^{5}a^{-5}+3z^{3}a^{-5}-za^{-5}+3z^{6}a^{-4}-3z^{4}a^{-4}-a^{-4}z^{-2}+3a^{-4}+a^{3}z^{7}+4z^{7}a^{-3}-5a^{3}z^{5}-10z^{5}a^{-3}+7a^{3}z^{3}+12z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-10a^{2}z^{6}-8z^{6}a^{-2}+16a^{2}z^{4}+12z^{4}a^{-2}-10a^{2}z^{2}-19z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+az^{9}+z^{9}a^{-1}-az^{7}+2z^{7}a^{-1}-9az^{5}-15z^{5}a^{-1}+16az^{3}+18z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+5z^{8}-21z^{6}+31z^{4}-26z^{2}-z^{-2}+11}$ (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         22
9        31-2
7       31 2
5      33  0
3     53   2
1    46    2
-1   22     0
-3  14      3
-5 12       -1
-7 1        1
-91         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

### Computer Talk

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