L10n44

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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_5,13,6,12 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 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 $u$ , $v$ , $w$ , ...)	${\frac {(u+v-1)(uv-u-v)}{uv}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+a^{5}z^{3}+a^{5}z+a^{3}z^{3}+a^{3}z+a^{3}z^{-1}-2az-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+3a^{8}z^{2}+2a^{7}z^{7}-9a^{7}z^{5}+10a^{7}z^{3}-3a^{7}z+a^{6}z^{8}-3a^{6}z^{6}+a^{6}z^{2}+3a^{5}z^{7}-13a^{5}z^{5}+15a^{5}z^{3}-5a^{5}z+a^{4}z^{8}-4a^{4}z^{6}+5a^{4}z^{4}-3a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-a^{3}z^{-1}+a^{2}z^{4}-a^{2}z^{2}+a^{2}+2az-az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

χ

0

2

-2

1

0

-4

2

1

-6

2

0

-8

1

0

-10

1

2

1

-12

1

0

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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