L10n37

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_17,5,18,20 X_7,19,8,18 X_19,9,20,8 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, -4, 5, 10, -2, -3, 4, -5, 3, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(v^{2}-v+1\right)\left(-uv^{2}+uv+u+v^{3}+v^{2}-v\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+3q^{7/2}-2q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}-q^{13/2}+2q^{11/2}-2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+z^{5}a^{-3}+az^{3}-7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+4az-13za^{-1}+9za^{-3}-2za^{-5}+4az^{-1}-8a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+2z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-2a^{-6}+z^{7}a^{-5}-2z^{5}a^{-5}-2z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}+3z^{6}a^{-4}-13z^{4}a^{-4}+18z^{2}a^{-4}-9a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-8za^{-3}+5a^{-3}z^{-1}+2z^{6}a^{-2}-14z^{4}a^{-2}+28z^{2}a^{-2}-14a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-9z^{5}a^{-1}+15az^{3}+23z^{3}a^{-1}-13az-21za^{-1}+4az^{-1}+8a^{-1}z^{-1}+z^{6}-8z^{4}+15z^{2}-8$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

1

-1

10

1

0

8

2

1

-1

6

1

-1

4

2

0

2

1

2

1

0

3

-2

1

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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