# L11n328

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 Planar diagram presentation X6172 X5,14,6,15 X8493 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 X4,17,1,18 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 ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-1)(w-1)(vw+1)}{{\sqrt {u}}vw}}}$ (db) Jones polynomial ${\displaystyle q^{5}-2q^{4}-q^{-4}+4q^{3}+2q^{-3}-4q^{2}-3q^{-2}+6q+5q^{-1}-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-3a^{2}z^{2}-7z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-7a^{-2}+z^{6}+5z^{4}+9z^{2}+z^{-2}+7}$ (db) Kauffman polynomial ${\displaystyle az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-z^{7}a^{-1}+2z^{7}a^{-3}-10a^{2}z^{6}-7z^{6}a^{-2}+z^{6}a^{-4}-18z^{6}-5a^{3}z^{5}-6az^{5}-8z^{5}a^{-1}-7z^{5}a^{-3}+15a^{2}z^{4}+9z^{4}a^{-2}-2z^{4}a^{-4}+26z^{4}+7a^{3}z^{3}+13az^{3}+14z^{3}a^{-1}+10z^{3}a^{-3}+2z^{3}a^{-5}-9a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}a^{-6}-21z^{2}-2a^{3}z-7az-10za^{-1}-6za^{-3}-za^{-5}+3a^{2}+7a^{-2}+a^{-4}-a^{-6}+9+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}}$ (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 \
-5-4-3-2-101234χ
11         11
9        1 -1
7       31 2
5      33  0
3     321  2
1    35    2
-1   221    1
-3  13      2
-5 12       -1
-7 1        1
-91         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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