L10n21

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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_9,16,10,17 X_17,5,18,20 X_13,19,14,18 X_19,15,20,14 X_15,8,16,9 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 $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)^{3}}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+5q^{7/2}-6q^{5/2}+5q^{3/2}-{\frac {1}{q^{3/2}}}-q^{13/2}+3q^{11/2}-5{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-2z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}+az-3za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-8z^{4}a^{-6}+3z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-2za^{-5}+z^{8}a^{-4}+z^{6}a^{-4}-7z^{4}a^{-4}+5z^{2}a^{-4}+4z^{7}a^{-3}-12z^{5}a^{-3}+15z^{3}a^{-3}-6za^{-3}+z^{8}a^{-2}-2z^{6}a^{-2}+3z^{4}a^{-2}+z^{2}a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+az^{3}+9z^{3}a^{-1}-2az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{4}-z^{2}-1$ (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

6

χ

14

1

12

2

-2

10

2

1

8

3

2

-1

6

3

2

1

4

2

3

1

2

3

0

1

4

3

-2

1

-1

-4

1

Integral Khovanov Homology

(db, data source)

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