L10n32

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, -7, 6, -5, -2, 10, 7, -8, 3, -4, 5, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-2a^{3}z^{-1}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+az^{-1}+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-2a^{3}z^{7}-az^{7}+6a^{4}z^{6}+7a^{2}z^{6}-z^{6}a^{-2}+6a^{5}z^{5}+13a^{3}z^{5}+7az^{5}-z^{5}a^{-1}-z^{5}a^{-3}-10a^{4}z^{4}-14a^{2}z^{4}+4z^{4}a^{-2}-10a^{5}z^{3}-23a^{3}z^{3}-12az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+6a^{4}z^{2}+11a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+5a^{5}z+13a^{3}z+6az-5za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

0

4

1

0

2

1

0

2

1

-2

1

2

1

-4

1

-1

-6

1

0

-8

1

0

-10

0

-12

1

Integral Khovanov Homology

(db, data source)

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