L10n70

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_11,16,12,17 X_17,20,18,11 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10}, {-7, 2, 4, -5, 9, 7, -8, 6, -10, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+2q^{-7}-2q^{-6}+3q^{-5}-2q^{-4}+3q^{-3}+q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{2}a^{4}-2a^{4}z^{-2}-a^{4}+z^{2}a^{2}+a^{2}z^{-2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+3a^{10}z^{2}-a^{10}+2a^{9}z^{7}-9a^{9}z^{5}+9a^{9}z^{3}-2a^{9}z+a^{8}z^{8}-3a^{8}z^{6}-2a^{8}z^{4}+6a^{8}z^{2}-a^{8}+3a^{7}z^{7}-14a^{7}z^{5}+17a^{7}z^{3}-6a^{7}z+a^{6}z^{8}-4a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}+a^{6}z^{-2}-a^{6}+a^{5}z^{7}-5a^{5}z^{5}+8a^{5}z^{3}-2a^{5}z-2a^{5}z^{-1}+a^{4}z^{4}-a^{4}z^{2}+2a^{4}z^{-2}-2a^{4}+2a^{3}z-2a^{3}z^{-1}+a^{2}z^{2}+a^{2}z^{-2}-2a^{2}$ (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

-1

1

-3

1

2

1

-5

2

1

3

-7

1

2

1

-9

3

2

1

-11

1

3

1

-13

1

2

1

0

-15

1

0

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

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