L10n74

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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_4,17,1,18 X_19,12,20,5 X_11,20,12,13 X_13,10,14,11
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w^{2}+uv^{2}w^{3}-uv^{2}w^{2}+vw-v-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+2q^{-5}-q^{-6}+2q^{-7}-2q^{-8}+2q^{-9}+q^{-11}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-z^{4}a^{10}-5z^{2}a^{10}-2a^{10}z^{-2}-5a^{10}+z^{6}a^{8}+5z^{4}a^{8}+6z^{2}a^{8}+a^{8}z^{-2}+2a^{8}+z^{6}a^{6}+5z^{4}a^{6}+6z^{2}a^{6}+2a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{2}-2a^{14}+a^{13}z+a^{12}z^{4}-3a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{7}-6a^{11}z^{5}+12a^{11}z^{3}-8a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-6a^{10}z^{6}+13a^{10}z^{4}-15a^{10}z^{2}-2a^{10}z^{-2}+9a^{10}+2a^{9}z^{7}-10a^{9}z^{5}+14a^{9}z^{3}-8a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-5a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+a^{7}z^{7}-4a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-2a^{6}$ (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

-5

1

-7

1

0

-9

1

-11

1

-13

1

3

1

-15

1

0

-17

2

0

-19

1

2

-21

2

1

-23

1

Integral Khovanov Homology

(db, data source)

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