L10n42

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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_15,5,16,4 X_5,13,6,12 X_16,12,17,11 X_6,18,1,17 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 $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-u^{2}v^{3}-uv^{4}+uv^{2}-u-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{9/2}+q^{7/2}-2q^{5/2}+q^{3/2}+q^{15/2}-q^{13/2}+q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+5za^{-7}+a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-9za^{-5}-3a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+5z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}+6z^{5}a^{-3}+17z^{5}a^{-5}+11z^{5}a^{-7}-5z^{4}a^{-4}+5z^{4}a^{-8}-11z^{3}a^{-3}-27z^{3}a^{-5}-16z^{3}a^{-7}-2z^{2}a^{-4}-7z^{2}a^{-6}-5z^{2}a^{-8}+8za^{-3}+15za^{-5}+8za^{-7}+za^{-9}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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

0

12

1

0

10

1

8

1

2

1

6

1

4

1

2

1

2

0

1

Integral Khovanov Homology

(db, data source)

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