L10n53

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,7,14 X_9,15,10,14 X_19,11,20,10 X_5,16,6,17 X_15,18,16,19 X₂₇₃₈ X_4,11,5,12 X_17,6,18,1
Gauss code	{1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

χ

2

1

0

2

-2

3

1

2

-4

4

3

-1

-6

4

2

-8

3

4

1

-10

3

4

-1

-12

1

3

2

-14

1

3

-2

-16

2

Integral Khovanov Homology

(db, data source)

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