L10n50

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_13,20,14,7 X_12,5,13,6 X_10,4,11,3 X_4,15,5,16 X_16,12,17,11 X_19,14,20,15 X_2,18,3,17
Gauss code	{1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, -4, 9, 7, -8, 10, -2, -9, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+a^{5}z^{5}+4a^{5}z^{3}+3a^{5}z-a^{3}z^{7}-5a^{3}z^{5}-7a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{5}+3az^{3}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+3a^{7}z^{3}-2a^{7}z+a^{6}z^{6}+3a^{5}z^{7}-11a^{5}z^{5}+15a^{5}z^{3}-6a^{5}z+2a^{4}z^{8}-6a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+6a^{3}z^{7}-23a^{3}z^{5}+23a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+2a^{2}z^{8}-6a^{2}z^{6}+2a^{2}z^{4}+a^{2}+3az^{7}-12az^{5}+11az^{3}-az-az^{-1}+z^{6}-3z^{4}+z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

1

0

-2

4

2

-4

2

0

-6

3

0

-8

2

3

1

-10

1

2

-1

-12

2

-14

1

-1

Integral Khovanov Homology

(db, data source)

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