L11n118

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 Planar diagram presentation X6172 X3,15,4,14 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 X2536 X13,1,14,4 Gauss code {1, -10, -2, 11}, {10, -1, -6, 5, -3, 8, -4, 7, -11, 2, -8, 3, -9, 6, -7, 4, -5, 9}
A Braid Representative

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -10q^{9/2}+11q^{7/2}-13q^{5/2}+11q^{3/2}-{\frac {2}{q^{3/2}}}+q^{15/2}-3q^{13/2}+7q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}}$ (db) Signature 1 (db) HOMFLY-PT polynomial ${\displaystyle 2z^{5}a^{-3}-4z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}+2az-6za^{-1}+8za^{-3}-5za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2z^{9}a^{-3}-2z^{9}a^{-5}-6z^{8}a^{-2}-10z^{8}a^{-4}-4z^{8}a^{-6}-5z^{7}a^{-1}-6z^{7}a^{-3}-4z^{7}a^{-5}-3z^{7}a^{-7}+17z^{6}a^{-2}+27z^{6}a^{-4}+8z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+13z^{5}a^{-1}+30z^{5}a^{-3}+25z^{5}a^{-5}+8z^{5}a^{-7}-24z^{4}a^{-2}-22z^{4}a^{-4}+z^{4}a^{-6}+3z^{4}a^{-8}-4z^{4}-3az^{3}-23z^{3}a^{-1}-39z^{3}a^{-3}-25z^{3}a^{-5}-6z^{3}a^{-7}+11z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}a^{-8}+4z^{2}+4az+13za^{-1}+18za^{-3}+12za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}}$ (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 \
-2-101234567χ
16         1-1
14        2 2
12       51 -4
10      52  3
8     65   -1
6    75    2
4   46     2
2  57      -2
0 26       4
-2 3        -3
-42         2
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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