# L11n244

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 Planar diagram presentation X12,1,13,2 X10,11,1,12 X5,15,6,14 X9,19,10,18 X17,3,18,2 X7,16,8,17 X3849 X15,20,16,21 X22,13,11,14 X19,4,20,5 X21,7,22,6 Gauss code {1, 5, -7, 10, -3, 11, -6, 7, -4, -2}, {2, -1, 9, 3, -8, 6, -5, 4, -10, 8, -11, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle 0}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-2{\sqrt {q}}}$ (db) Signature -2 (db) HOMFLY-PT polynomial ${\displaystyle z^{3}a^{5}+2za^{5}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}+z^{5}a+5z^{3}a+6za+az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}-a^{5}z^{7}+4a^{3}z^{7}+4az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+5a^{4}z^{6}+18a^{2}z^{6}+11z^{6}-a^{7}z^{5}+5a^{5}z^{5}+5a^{3}z^{5}+4az^{5}+5z^{5}a^{-1}+8a^{6}z^{4}-2a^{4}z^{4}-24a^{2}z^{4}-14z^{4}+3a^{7}z^{3}-5a^{5}z^{3}-21a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-4a^{6}z^{2}-4a^{4}z^{2}+4a^{2}z^{2}+4z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-6-5-4-3-2-101234χ
6          11
4         1 -1
2        11 0
0      321  2
-2     131   1
-4    222    2
-6   121     0
-8  121      0
-10 111       -1
-12 1         -1
-141          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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