# L10a72

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 Planar diagram presentation X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X20,13,7,14 X12,19,13,20 X2738 X6,9,1,10 Gauss code {1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -6, 8, -7}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {2t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}-2t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-4t(1)t(2)+2t(2)-t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {5}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {10}{q^{17/2}}}+{\frac {10}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {7}{q^{11/2}}}}$ (db) Signature -5 (db) HOMFLY-PT polynomial ${\displaystyle a^{11}z^{3}+a^{11}z-a^{11}z^{-1}-a^{9}z^{5}-a^{9}z^{3}+3a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-6a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-2a^{5}z}$ (db) Kauffman polynomial ${\displaystyle -z^{5}a^{15}+2z^{3}a^{15}-3z^{6}a^{14}+7z^{4}a^{14}-3z^{2}a^{14}-4z^{7}a^{13}+9z^{5}a^{13}-6z^{3}a^{13}+2za^{13}-3z^{8}a^{12}+4z^{6}a^{12}-2z^{2}a^{12}+a^{12}-z^{9}a^{11}-4z^{7}a^{11}+12z^{5}a^{11}-13z^{3}a^{11}+5za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+9z^{6}a^{10}-5z^{4}a^{10}-5z^{2}a^{10}+3a^{10}-z^{9}a^{9}-3z^{7}a^{9}+10z^{5}a^{9}-16z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{8}a^{8}+6z^{4}a^{8}-7z^{2}a^{8}+3a^{8}-3z^{7}a^{7}+7z^{5}a^{7}-8z^{3}a^{7}+7za^{7}-2a^{7}z^{-1}-2z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-2za^{5}}$ (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
-4          11
-6         21-1
-8        3  3
-10       42  -2
-12      63   3
-14     44    0
-16    66     0
-18   35      2
-20  25       -3
-22 13        2
-24 2         -2
-261          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-6}$ ${\displaystyle i=-4}$ ${\displaystyle r=-10}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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