L10n52

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X_14,6,15,5 X_16,12,17,11 X_3,10,4,11 X_12,7,13,8 X_20,15,7,16 X_6,14,1,13 X_4,19,5,20 X_17,2,18,3
Gauss code	{1, 10, -5, -9, 3, -8}, {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-u^{2}v^{2}+2u^{2}v-u^{2}+uv^{4}-3uv^{3}+5uv^{2}-3uv+u-v^{4}+2v^{3}-v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{3/2}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}+3z^{3}a+5za+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{8}-2a^{2}z^{8}-4a^{5}z^{7}-7a^{3}z^{7}-3az^{7}-3a^{6}z^{6}+2a^{4}z^{6}+4a^{2}z^{6}-z^{6}-a^{7}z^{5}+11a^{5}z^{5}+21a^{3}z^{5}+9az^{5}+7a^{6}z^{4}+2a^{4}z^{4}-7a^{2}z^{4}-2z^{4}+2a^{7}z^{3}-9a^{5}z^{3}-26a^{3}z^{3}-18az^{3}-3z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+4a^{2}z^{2}+3z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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

2

-2

0

4

2

-2

4

3

-1

-4

4

3

1

-6

3

5

2

-8

2

3

-1

-10

1

3

2

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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