L10n43

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_16,12,17,11 X_17,6,18,1 X_19,15,20,14 X_13,19,14,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 10, -9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

2

1

8

4

2

-2

6

3

2

1

4

3

4

1

2

3

0

1

4

3

-2

1

2

-1

-4

2

Integral Khovanov Homology

(db, data source)

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