L10a143

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_12,15,5,16 X_18,9,19,10 X_16,7,17,8 X_20,11,13,12 X_10,17,11,18 X_8,19,9,20 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -8, 4, -7, 6, -3}, {10, -2, 3, -5, 7, -4, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w+2uv^{2}w^{2}-3uv^{2}w+uv^{2}+uvw^{3}-3uvw^{2}+3uvw-uv+uw^{2}-uw+v^{3}w^{2}-v^{3}w+v^{2}w^{3}-3v^{2}w^{2}+3v^{2}w-v^{2}-vw^{3}+3vw^{2}-2vw-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-9q^{-5}+11q^{-6}-11q^{-7}+11q^{-8}-7q^{-9}+5q^{-10}-2q^{-11}+q^{-12}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-3a^{10}z^{2}-2a^{10}z^{-2}-5a^{10}+2a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}+2a^{8}+3a^{6}z^{4}+6a^{6}z^{2}+2a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{14}z^{6}-4a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+2a^{13}z^{7}-6a^{13}z^{5}+4a^{13}z^{3}+a^{13}z+2a^{12}z^{8}-3a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{9}+3a^{11}z^{7}-12a^{11}z^{5}+12a^{11}z^{3}-8a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-13a^{10}z^{6}+14a^{10}z^{4}-16a^{10}z^{2}-2a^{10}z^{-2}+9a^{10}+a^{9}z^{9}+7a^{9}z^{7}-16a^{9}z^{5}+12a^{9}z^{3}-8a^{9}z+2a^{9}z^{-1}+4a^{8}z^{8}-3a^{8}z^{6}+2a^{8}z^{4}-4a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+6a^{7}z^{7}-7a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+6a^{6}z^{6}-9a^{6}z^{4}+7a^{6}z^{2}-2a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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

-3

1

-5

3

1

-2

-7

4

-9

5

3

-2

-11

6

4

2

-13

6

0

-15

5

0

-17

2

6

4

-19

3

5

-2

-21

1

4

3

-23

1

-1

-25

1

Integral Khovanov Homology

(db, data source)

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