L10n47

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_16,12,17,11 X_14,20,15,19 X_17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, 4, -9, 7, -8, -10, 2, 9, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-u^{2}v^{2}+2u^{2}v+uv^{4}-2uv^{3}+3uv^{2}-2uv+u+2v^{3}-v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-q^{3/2}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-7za^{3}-3a^{3}z^{-1}+2z^{3}a+3za+az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-3a^{5}z^{7}-4a^{3}z^{7}-az^{7}-2a^{6}z^{6}+2a^{2}z^{6}-a^{7}z^{5}+11a^{5}z^{5}+14a^{3}z^{5}+2az^{5}+5a^{6}z^{4}+6a^{4}z^{4}-2a^{2}z^{4}-3z^{4}+3a^{7}z^{3}-14a^{5}z^{3}-21a^{3}z^{3}-5az^{3}-z^{3}a^{-1}-a^{6}z^{2}-8a^{4}z^{2}-4a^{2}z^{2}+3z^{2}-a^{7}z+9a^{5}z+12a^{3}z+3az+za^{-1}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

-2

0

2

1

-2

4

3

-1

-4

2

1

-6

2

4

2

-8

2

0

-10

2

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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