L10n82

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,20,12,13 X_7,18,8,19 X_9,16,10,17 X_17,8,18,9 X_13,10,14,11 X_19,12,20,5 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{2}-uv^{2}w-uvw^{2}+uvw+uw^{3}-v^{3}-v^{2}w^{2}+v^{2}w+vw^{2}-vw}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-8}+q^{-6}+q^{-5}-2q^{-4}+3q^{-3}-2q^{-2}+q+3q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-a^{6}z^{2}-2a^{6}z^{-2}-3a^{6}+a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{4}-2a^{2}z^{2}+z^{2}+1$ (db)
Kauffman polynomial	$a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-22a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+a^{7}z^{7}-8a^{7}z^{5}+17a^{7}z^{3}-11a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-9a^{6}z^{6}+25a^{6}z^{4}-27a^{6}z^{2}-2a^{6}z^{-2}+13a^{6}+a^{5}z^{7}-8a^{5}z^{5}+19a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{4}-6a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+2a^{3}z^{5}-2a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{2}+2az^{5}-4az^{3}+z^{4}-2z^{2}+1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

2

1

-3

2

3

1

-5

1

2

1

-7

1

2

1

-9

1

2

-1

-11

2

-13

1

-15

1

-17

1

Integral Khovanov Homology

(db, data source)

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