L10a72

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,17,15,18 X_16,5,17,6 X_4,15,5,16 X_18,11,19,12 X_20,13,7,14 X_12,19,13,20 X₂₇₃₈ X_6,9,1,10
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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}-2t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-4t(1)t(2)+2t(2)-t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {5}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {10}{q^{17/2}}}+{\frac {10}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {7}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+a^{11}z-a^{11}z^{-1}-a^{9}z^{5}-a^{9}z^{3}+3a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-6a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-2a^{5}z$ (db)
Kauffman polynomial	$-z^{5}a^{15}+2z^{3}a^{15}-3z^{6}a^{14}+7z^{4}a^{14}-3z^{2}a^{14}-4z^{7}a^{13}+9z^{5}a^{13}-6z^{3}a^{13}+2za^{13}-3z^{8}a^{12}+4z^{6}a^{12}-2z^{2}a^{12}+a^{12}-z^{9}a^{11}-4z^{7}a^{11}+12z^{5}a^{11}-13z^{3}a^{11}+5za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+9z^{6}a^{10}-5z^{4}a^{10}-5z^{2}a^{10}+3a^{10}-z^{9}a^{9}-3z^{7}a^{9}+10z^{5}a^{9}-16z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{8}a^{8}+6z^{4}a^{8}-7z^{2}a^{8}+3a^{8}-3z^{7}a^{7}+7z^{5}a^{7}-8z^{3}a^{7}+7za^{7}-2a^{7}z^{-1}-2z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-2za^{5}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

2

1

-1

-8

3

-10

4

2

-2

-12

6

3

-14

4

0

-16

6

0

-18

3

5

2

-20

2

5

-3

-22

1

3

2

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$