# L11n321

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+2uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w-2v^{2}w^{2}+4v^{2}w-v^{2}+vw^{2}-2vw+v}{{\sqrt {u}}v^{3/2}w^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -2q^{3}+5q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+4q^{-4}-2q^{-5}+q^{-6}}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle -a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+3z^{4}+3a^{4}z^{2}-11a^{2}z^{2}-2z^{2}a^{-2}+9z^{2}+3a^{4}-9a^{2}-2a^{-2}+8+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}}$ (db) Kauffman polynomial ${\displaystyle a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}-a^{6}+2a^{5}z^{7}-7a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+2a^{4}z^{8}-5a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+a^{3}z^{9}+a^{3}z^{7}-10a^{3}z^{5}+13a^{3}z^{3}+3z^{3}a^{-3}-8a^{3}z-2za^{-3}+2a^{3}z^{-1}+5a^{2}z^{8}-16a^{2}z^{6}+z^{6}a^{-2}+24a^{2}z^{4}+4z^{4}a^{-2}-23a^{2}z^{2}-7z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}+3a^{-2}+az^{9}+2az^{7}+3z^{7}a^{-1}-9az^{5}-6z^{5}a^{-1}+15az^{3}+11z^{3}a^{-1}-12az-7za^{-1}+2az^{-1}+3z^{8}-9z^{6}+22z^{4}-24z^{2}-z^{-2}+11}$ (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 \
-6-5-4-3-2-10123χ
7         2-2
5        3 3
3       32 -1
1      63  3
-1     56   1
-3    43    1
-5   25     3
-7  24      -2
-9 13       2
-11 1        -1
-131         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

### Computer Talk

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