L10n75

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_17,1,18,4 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

A Morse Link Presentation

Polynomial invariants

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

2

-3

3

2

-1

-5

2

-7

2

3

1

-9

3

2

1

-11

2

3

1

-13

1

2

-1

-15

2

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

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