L10n56

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_18,11,19,12 X_20,5,9,6 X_7,15,8,14 X_12,4,13,3 X_13,16,14,17 X_15,7,16,6 X_8,9,1,10 X_4,19,5,20 X_2,18,3,17
Gauss code	{1, -10, 5, -9, 3, 7, -4, -8}, {8, -1, 2, -5, -6, 4, -7, 6, 10, -2, 9, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(u+1)^{2}(v-1)}{u^{3/2}{\sqrt {v}}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{5}(-z)-a^{3}z+az^{5}+5az^{3}-z^{3}a^{-1}+5az+az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}-a^{3}z^{7}-2az^{7}-z^{7}a^{-1}+6a^{2}z^{6}+6z^{6}-a^{5}z^{5}+6a^{3}z^{5}+13az^{5}+6z^{5}a^{-1}-a^{6}z^{4}-8a^{2}z^{4}-9z^{4}+4a^{5}z^{3}-8a^{3}z^{3}-22az^{3}-10z^{3}a^{-1}+3a^{6}z^{2}+a^{4}z^{2}+a^{2}z^{2}+3z^{2}-2a^{5}z+2a^{3}z+10az+6za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

0

2

1

0

2

1

-2

1

2

1

0

-4

1

-6

1

-8

1

-10

0

-12

1

-1

Integral Khovanov Homology

(db, data source)

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