L10n79

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_11,20,12,13 X_13,12,14,5 X_4,17,1,18 X_19,10,20,11
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, {-8, 2, 4, -5, 9, 6, -10, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

2

0

-7

3

-9

2

0

-11

4

3

1

-13

1

2

1

-15

3

4

-1

-17

1

3

2

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{2}$