L10n19

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_11,14,12,15 X₃₈₄₉ X_5,13,6,12 X_13,5,14,20 X_9,16,10,17 X_15,10,16,11 X_17,2,18,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -8, 9, -4, 6, -7, 4, -9, 8, -10, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{3}-4t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-4t(2)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-3a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+a^{8}z^{6}-2a^{8}z^{2}+a^{8}+2a^{7}z^{7}-8a^{7}z^{5}+18a^{7}z^{3}-13a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-3a^{6}z^{6}+8a^{6}z^{4}-7a^{6}z^{2}+3a^{6}+3a^{5}z^{7}-10a^{5}z^{5}+16a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}+6a^{4}z^{4}-6a^{4}z^{2}+2a^{4}+a^{3}z^{7}-a^{3}z^{5}-3a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+a^{2}z^{6}-2a^{2}z^{4}-a^{2}z^{2}+a^{2}+az^{5}-4az^{3}+4az-az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

-2

3

1

2

-4

2

1

-1

-6

3

2

1

-8

2

0

-10

2

3

-1

-12

2

3

1

-14

1

-1

-16

2

Integral Khovanov Homology

(db, data source)

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