L10n54

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_17,5,18,4 X_5,19,6,18 X_6,14,7,13 X_14,8,15,7 X_8,16,1,15 X_19,17,20,16
Gauss code	{1, -4, 2, 5, -6, -7, 8, -9}, {4, -1, 3, -2, 7, -8, 9, 10, -5, 6, -10, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}-t(1)-t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+2q^{7/2}-2q^{5/2}+q^{3/2}+q^{15/2}-2q^{13/2}+q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+5za^{-7}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-8za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+2z^{2}a^{-8}+2z^{7}a^{-7}-10z^{5}a^{-7}+12z^{3}a^{-7}-3za^{-7}+z^{8}a^{-6}-4z^{6}a^{-6}+2z^{4}a^{-6}+z^{2}a^{-6}+3z^{7}a^{-5}-16z^{5}a^{-5}+23z^{3}a^{-5}-10za^{-5}+a^{-5}z^{-1}+z^{8}a^{-4}-5z^{6}a^{-4}+6z^{4}a^{-4}-z^{2}a^{-4}-a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+11z^{3}a^{-3}-7za^{-3}+a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

χ

16

1

-1

14

1

12

1

2

1

10

1

8

1

0

6

1

0

4

1

2

1

2

0

1

Integral Khovanov Homology

(db, data source)

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