L10n81

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_3,10,4,11 X_5,14,6,15 X_11,20,12,13 X_13,12,14,5 X_19,1,20,4 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+t(2)-t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-6q^{-7}+6q^{-6}-6q^{-5}+6q^{-4}-2q^{-3}+2q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{6}-5a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-11a^{6}+2a^{4}z^{4}+7a^{4}z^{2}+a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-2a^{11}z^{3}+3a^{10}z^{6}-4a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{7}-2a^{9}z^{3}+a^{8}z^{8}+4a^{8}z^{4}-7a^{8}z^{2}-a^{8}z^{-2}+5a^{8}+3a^{7}z^{7}-5a^{7}z^{5}+9a^{7}z^{3}-11a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-3a^{6}z^{6}+12a^{6}z^{4}-20a^{6}z^{2}-2a^{6}z^{-2}+13a^{6}+a^{5}z^{7}-3a^{5}z^{5}+9a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{4}-9a^{4}z^{2}-a^{4}z^{-2}+8a^{4}$ (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

-3

2

-5

2

0

-7

4

-9

2

0

-11

4

0

-13

3

0

-15

2

3

-1

-17

3

-19

1

2

-1

-21

1

Integral Khovanov Homology

(db, data source)

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