L10n93

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,18,10,19 X_17,6,18,1 X_16,7,17,8 X_3,10,4,11 X_5,14,6,15 X_13,4,14,5 X_20,11,13,12 X_12,15,7,16 X_2,19,3,20
Gauss code	{1, -10, -5, 7, -6, 3}, {4, -1, -2, 5, 8, -9}, {-7, 6, 9, -4, -3, 2, 10, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(uvw+1)\left(uvw^{2}-1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-11}+q^{-9}+q^{-6}+q^{-4}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{2}a^{12}+a^{12}z^{-2}+2a^{12}-z^{6}a^{10}-7z^{4}a^{10}-14z^{2}a^{10}-2a^{10}z^{-2}-9a^{10}+z^{8}a^{8}+8z^{6}a^{8}+21z^{4}a^{8}+21z^{2}a^{8}+a^{8}z^{-2}+7a^{8}$ (db)
Kauffman polynomial	$a^{14}z^{2}-2a^{14}-a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{7}-7a^{11}z^{5}+14a^{11}z^{3}-9a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+21a^{10}z^{4}-23a^{10}z^{2}-2a^{10}z^{-2}+11a^{10}+a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-9a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}-a^{8}z^{-2}+7a^{8}$ (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

1

0

-17

2

1

-19

2

1

-21

1

-23

1

Integral Khovanov Homology

(db, data source)

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