L10n51

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_11,19,12,18 X_5,12,6,13 X_17,4,18,5 X_14,7,15,8 X_16,14,17,13 X_20,15,7,16 X_6,19,1,20
Gauss code	{1, -2, 3, 6, -5, -10}, {7, -1, 2, -3, -4, 5, 8, -7, 9, -8, -6, 4, 10, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2u^{2}v^{2}+u^{2}v+uv^{4}-3uv^{3}+3uv^{2}-3uv+u+v^{3}-2v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-2a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-2z^{7}a^{9}+4z^{5}a^{9}-2z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-z^{8}a^{8}+2z^{4}a^{8}-4z^{7}a^{7}+11z^{5}a^{7}-15z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-z^{8}a^{6}+z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}-2z^{7}a^{5}+6z^{5}a^{5}-13z^{3}a^{5}+9za^{5}-2a^{5}z^{-1}-z^{6}a^{4}-3z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

1

-1

-6

3

1

2

-8

3

2

-1

-10

3

0

-12

2

3

1

-14

2

3

-1

-16

1

3

2

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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