L10n64

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X₃₉₄₈ X_17,2,18,3 X_14,6,15,5 X_6,12,7,11 X_9,18,10,19 X_20,15,11,16 X_10,13,1,14 X_4,19,5,20
Gauss code	{1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}\left(-v^{2}\right)+2u^{3}v-u^{3}+u^{2}v^{2}-3u^{2}v+u^{2}+uv^{3}-3uv^{2}+uv-v^{3}+2v^{2}-v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+2a^{5}z^{3}+3a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}+az^{3}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-11a^{7}z^{5}+10a^{7}z^{3}-2a^{7}z+2a^{6}z^{8}-4a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}+7a^{5}z^{7}-24a^{5}z^{5}+23a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}-3a^{4}z^{6}-2a^{4}z^{4}+2a^{4}z^{2}-a^{4}+4a^{3}z^{7}-13a^{3}z^{5}+17a^{3}z^{3}-8a^{3}z+a^{3}z^{-1}+2a^{2}z^{6}-2a^{2}z^{4}+2a^{2}z^{2}+4az^{3}-az+z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

3

-2

3

2

-1

-4

3

2

1

-6

3

0

-8

3

0

-10

2

4

2

-12

1

2

-1

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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