L10n46

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X₆₇₁₈ X_13,20,14,7 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_16,12,17,11 X_19,14,20,15 X_17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, -4, 9, 7, -8, -10, 2, -9, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

χ

-2

2

-4

1

0

-6

2

1

-8

2

1

-1

-10

1

2

-1

-12

1

2

1

-14

1

0

-16

2

Integral Khovanov Homology

(db, data source)

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