L10n68

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_9,18,10,19 X_11,16,12,17 X_17,20,18,11 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)+t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+3q^{-5}-2q^{-6}+4q^{-7}-3q^{-8}+3q^{-9}-2q^{-10}+q^{-11}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}-a^{10}z^{4}-4a^{10}z^{2}+a^{10}z^{-2}-2a^{10}+a^{8}z^{6}+4a^{8}z^{4}+2a^{8}z^{2}-2a^{8}z^{-2}-3a^{8}+a^{6}z^{6}+5a^{6}z^{4}+7a^{6}z^{2}+a^{6}z^{-2}+4a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{2}-a^{14}+2a^{13}z^{3}-2a^{13}z+a^{12}z^{6}-3a^{12}z^{4}+5a^{12}z^{2}-a^{12}+2a^{11}z^{7}-9a^{11}z^{5}+14a^{11}z^{3}-6a^{11}z+a^{10}z^{8}-3a^{10}z^{6}+a^{10}z^{4}+3a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+3a^{9}z^{7}-12a^{9}z^{5}+11a^{9}z^{3}-2a^{9}z^{-1}+a^{8}z^{8}-3a^{8}z^{6}-a^{8}z^{4}+6a^{8}z^{2}+2a^{8}z^{-2}-6a^{8}+a^{7}z^{7}-3a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-5a^{6}z^{4}+7a^{6}z^{2}+a^{6}z^{-2}-4a^{6}$ (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

-5

1

-7

1

0

-9

2

-11

1

-13

4

2

-15

1

-17

3

0

-19

1

2

1

-21

1

-1

-23

1

Integral Khovanov Homology

(db, data source)

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