L10n66

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_16,12,17,11 X_18,13,19,14 X_20,18,9,17 X_12,19,13,20 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uvw^{2}+uvw+uw^{3}-2uw^{2}+2vw-v-w^{2}+w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+2q^{-4}+q^{3}-q^{-3}-q^{2}+q^{-2}+q+1$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2a^{4}z^{2}-3a^{4}z^{-2}-5a^{4}+a^{2}z^{4}+5a^{2}z^{2}+4a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+8a^{2}+2a^{-2}-z^{4}-5z^{2}-3z^{-2}-6$ (db)
Kauffman polynomial	$a^{5}z^{7}+a^{3}z^{7}+az^{7}+z^{7}a^{-1}+a^{6}z^{6}+3a^{4}z^{6}+3a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-4a^{5}z^{5}-5a^{3}z^{5}-6az^{5}-5z^{5}a^{-1}-5a^{6}z^{4}-16a^{4}z^{4}-20a^{2}z^{4}-5z^{4}a^{-2}-14z^{4}+2a^{5}z^{3}+4a^{3}z^{3}+6az^{3}+4z^{3}a^{-1}+7a^{6}z^{2}+23a^{4}z^{2}+35a^{2}z^{2}+6z^{2}a^{-2}+25z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

1

0

1

3

1

2

-1

1

3

1

-3

1

2

1

-5

1

0

-7

1

-9

1

2

1

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

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