# L10n84

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 Link L10n84. A graph, L10n84. A part of a link and a part of a graph.

 Planar diagram presentation X6172 X10,3,11,4 X13,20,14,15 X7,16,8,17 X15,8,16,9 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10 Gauss code {1, -9, 2, -10}, {-5, 4, -8, 6, -7, 3}, {9, -1, -4, 5, 10, -2, -6, 7, -3, 8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(w-1)\left(uv^{2}w^{2}+uv^{2}w+uv+vw^{3}+w^{2}+w\right)}{{\sqrt {u}}vw^{2}}}}$ (db) Jones polynomial ${\displaystyle q^{-12}-q^{-11}+2q^{-10}-q^{-9}+2q^{-8}-q^{-7}+q^{-6}+q^{-3}}$ (db) Signature -4 (db) HOMFLY-PT polynomial ${\displaystyle a^{12}z^{-2}+a^{12}-2z^{2}a^{10}-2a^{10}z^{-2}-4a^{10}-z^{2}a^{8}+a^{8}z^{-2}+z^{6}a^{6}+6z^{4}a^{6}+9z^{2}a^{6}+3a^{6}}$ (db) Kauffman polynomial ${\displaystyle a^{14}z^{6}-5a^{14}z^{4}+6a^{14}z^{2}-2a^{14}+a^{13}z^{7}-4a^{13}z^{5}+2a^{13}z^{3}+a^{13}z+a^{12}z^{8}-5a^{12}z^{6}+7a^{12}z^{4}-6a^{12}z^{2}-a^{12}z^{-2}+4a^{12}+2a^{11}z^{7}-11a^{11}z^{5}+17a^{11}z^{3}-10a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+17a^{10}z^{4}-20a^{10}z^{2}-2a^{10}z^{-2}+10a^{10}+a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}-a^{8}z^{4}+a^{8}z^{2}-a^{8}z^{-2}+2a^{8}-a^{7}z^{3}+a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+9a^{6}z^{2}-3a^{6}}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-5          11
-7          11
-9       11  0
-11      1    1
-13     231   0
-15    1      1
-17   131     1
-19  111      1
-21  1        1
-2311         0
-251          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-7}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-10}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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