L10n41

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_11,16,12,17 X_17,6,18,1 X_14,19,15,20 X_18,13,19,14
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}-2u^{2}v+u^{2}-uv^{2}+3uv-u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+2z^{3}a^{5}+5za^{5}+3a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}+z^{3}a+za$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}-a^{8}+2a^{7}z^{7}-8a^{7}z^{5}+8a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-a^{6}z^{6}-7a^{6}z^{4}+9a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-15a^{5}z^{5}+17a^{5}z^{3}-10a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}-a^{4}z^{6}-4a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}+a^{2}z^{6}-a^{2}z^{4}+2a^{2}z^{2}+3az^{3}-2az+z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

2

-2

2

0

-4

3

1

2

-6

2

3

1

-8

2

0

-10

1

2

1

-12

1

2

-1

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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