# L10a112

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 Planar diagram presentation X12,1,13,2 X8493 X14,6,15,5 X18,7,19,8 X16,10,17,9 X10,11,1,12 X6,14,7,13 X4,19,5,20 X20,15,11,16 X2,18,3,17 Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-v+1\right)}{u^{3/2}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{9/2}-{\frac {4}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+8q^{5/2}-{\frac {14}{q^{5/2}}}-13q^{3/2}+{\frac {16}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {18}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-2a^{3}z^{3}+7az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-2a^{3}z+5az-4za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{6}z^{4}+4a^{5}z^{5}-a^{5}z^{3}+9a^{4}z^{6}+z^{6}a^{-4}-8a^{4}z^{4}-2z^{4}a^{-4}+3a^{4}z^{2}+z^{2}a^{-4}+13a^{3}z^{7}+4z^{7}a^{-3}-20a^{3}z^{5}-10z^{5}a^{-3}+13a^{3}z^{3}+8z^{3}a^{-3}-4a^{3}z-2za^{-3}+10a^{2}z^{8}+6z^{8}a^{-2}-10a^{2}z^{6}-13z^{6}a^{-2}-3a^{2}z^{4}+6z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}+3az^{9}+3z^{9}a^{-1}+16az^{7}+7z^{7}a^{-1}-47az^{5}-33z^{5}a^{-1}+34az^{3}+28z^{3}a^{-1}-10az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+16z^{8}-33z^{6}+14z^{4}-1}$ (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 \
-5-4-3-2-1012345χ
10          1-1
8         3 3
6        51 -4
4       83  5
2      85   -3
0     108    2
-2    810     2
-4   68      -2
-6  38       5
-8 16        -5
-10 3         3
-121          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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