L10a73

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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_20,11,7,12 X_18,13,19,14 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, -7, 8, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

-2

1

-4

2

1

-1

-6

2

-8

3

2

-1

-10

4

2

-12

2

3

1

-14

4

0

-16

2

3

1

-18

1

3

-2

-20

1

2

1

-22

1

-1

-24

1

Integral Khovanov Homology

(db, data source)

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