L10n77

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_11,20,12,13 X_13,12,14,5 X_4,17,1,18 X_19,10,20,11
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, -4, -5, 9, 6, -10, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w^{3}-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-12}+q^{-8}+q^{-6}+q^{-4}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-a^{10}z^{6}-7a^{10}z^{4}-15a^{10}z^{2}-2a^{10}z^{-2}-11a^{10}+a^{8}z^{8}+8a^{8}z^{6}+21a^{8}z^{4}+21a^{8}z^{2}+a^{8}z^{-2}+8a^{8}$ (db)
Kauffman polynomial	$a^{16}+a^{12}z^{4}-5a^{12}z^{2}-a^{12}z^{-2}+5a^{12}+a^{11}z^{7}-7a^{11}z^{5}+15a^{11}z^{3}-11a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+22a^{10}z^{4}-26a^{10}z^{2}-2a^{10}z^{-2}+13a^{10}+a^{9}z^{7}-7a^{9}z^{5}+15a^{9}z^{3}-11a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}-a^{8}z^{-2}+8a^{8}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-7

1

-9

1

-11

1

-13

1

-15

2

1

-17

1

-19

1

0

-21

1

0

-23

2

1

-25

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-9$	$i=-7$	$i=-5$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$