L10n1

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{5}+2uv^{4}-2uv^{3}-2v^{2}+2v-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{5/2}}}+q^{3/2}-{\frac {3}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+5z^{3}a^{5}+7za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-6z^{5}a^{3}-12z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+z^{5}a+4z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+2a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{6}-4a^{6}z^{4}+8a^{6}z^{2}-4a^{6}+2a^{5}z^{7}-10a^{5}z^{5}+19a^{5}z^{3}-15a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}-2a^{4}z^{6}-6a^{4}z^{4}+15a^{4}z^{2}-7a^{4}+4a^{3}z^{7}-19a^{3}z^{5}+27a^{3}z^{3}-16a^{3}z+4a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}-6a^{2}z^{4}+11a^{2}z^{2}-4a^{2}+2az^{7}-9az^{5}+10az^{3}-4az+az^{-1}+z^{6}-4z^{4}+3z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

1

0

-2

1

3

1

-4

2

0

-6

3

2

1

-8

1

2

-10

2

0

-12

1

-14

1

-1

Integral Khovanov Homology

(db, data source)

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