L10n3

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_11,19,12,18 X_17,5,18,20 X_19,13,20,12 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -9, 4, -6, 8, 10, -2, 3, 9, -7, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {2}{q^{7/2}}}-2q^{5/2}+{\frac {3}{q^{5/2}}}+3q^{3/2}-{\frac {4}{q^{3/2}}}-5{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}-2a^{3}z+za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+az^{5}+4az^{3}-2z^{3}a^{-1}+6az-5za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}+z^{2}a^{-4}-a^{4}-a^{-4}+2a^{3}z^{7}-8a^{3}z^{5}+8a^{3}z^{3}+2z^{3}a^{-3}-4a^{3}z-2za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{8}-a^{2}z^{6}+z^{6}a^{-2}-8a^{2}z^{4}-3z^{4}a^{-2}+11a^{2}z^{2}+7z^{2}a^{-2}-4a^{2}-4a^{-2}+4az^{7}+2z^{7}a^{-1}-16az^{5}-8z^{5}a^{-1}+19az^{3}+13z^{3}a^{-1}-11az-9za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{8}-z^{6}-7z^{4}+13z^{2}-7$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

8

1

-1

6

1

4

2

1

-1

2

3

1

2

0

2

4

2

-2

2

1

-4

1

2

1

-6

1

2

-1

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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