L10n23

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_17,5,18,20 X_13,19,14,18 X_19,15,20,14 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$2q^{15/2}-2q^{13/2}+2q^{11/2}-3q^{9/2}+2q^{7/2}-3q^{5/2}+q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+6za^{-7}+2a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-12z^{3}a^{-5}-12za^{-5}-5a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+7za^{-3}+3a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{-10}+z^{3}a^{-9}-2za^{-9}+2z^{6}a^{-8}-8z^{4}a^{-8}+6z^{2}a^{-8}+3z^{7}a^{-7}-15z^{5}a^{-7}+21z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}-2z^{6}a^{-6}-7z^{4}a^{-6}+13z^{2}a^{-6}-5a^{-6}+4z^{7}a^{-5}-21z^{5}a^{-5}+32z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-4z^{6}a^{-4}+z^{4}a^{-4}+7z^{2}a^{-4}-5a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+12z^{3}a^{-3}-10za^{-3}+3a^{-3}z^{-1}$ (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

2

-2

14

1

0

12

2

0

10

1

8

1

2

1

6

2

1

4

1

3

2

0

1

Integral Khovanov Homology

(db, data source)

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