# L11n319

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 Planar diagram presentation X6172 X12,4,13,3 X14,5,15,6 X11,21,12,20 X17,11,18,22 X21,17,22,16 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15 Gauss code {1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {uv^{2}w^{4}-uv^{2}w^{3}-uvw^{4}+3uvw^{3}-2uvw^{2}+2uvw-uv-uw^{3}+uw^{2}-uw+u-v^{2}w^{4}+v^{2}w^{3}-v^{2}w^{2}+v^{2}w+vw^{4}-2vw^{3}+2vw^{2}-3vw+v+w-1}{{\sqrt {u}}vw^{2}}}}$ (db) Jones polynomial ${\displaystyle -2q^{6}+4q^{5}-7q^{4}+10q^{3}-9q^{2}+11q-7+6q^{-1}-3q^{-2}+q^{-3}}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle -z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-14z^{2}a^{-2}+9z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-10a^{-2}+8a^{-4}-2a^{-6}+4-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}}$ (db) Kauffman polynomial ${\displaystyle 2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}-z^{7}a^{-1}+4z^{7}a^{-5}+a^{2}z^{6}-34z^{6}a^{-2}-19z^{6}a^{-4}+z^{6}a^{-6}-13z^{6}-9az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-13z^{5}a^{-5}-3a^{2}z^{4}+50z^{4}a^{-2}+35z^{4}a^{-4}+2z^{4}a^{-6}+14z^{4}+5az^{3}+14z^{3}a^{-1}+31z^{3}a^{-3}+25z^{3}a^{-5}+3z^{3}a^{-7}+2a^{2}z^{2}-37z^{2}a^{-2}-26z^{2}a^{-4}-4z^{2}a^{-6}-13z^{2}-11za^{-1}-24za^{-3}-18za^{-5}-5za^{-7}+15a^{-2}+12a^{-4}+3a^{-6}+7+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}}$ (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 \
-4-3-2-1012345χ
13         2-2
11        2 2
9       52 -3
7      52  3
5     56   1
3    64    2
1   37     4
-1  34      -1
-3 14       3
-5 2        -2
-71         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

### Computer Talk

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