L10n31

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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 X_3,10,4,11 X_5,13,6,12 X_13,5,14,20 X_16,9,17,10 X_15,2,16,3 X_8,17,9,18
Gauss code	{1, 9, -5, -3}, {-6, -1, 2, -10, 8, 5, -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 $u$ , $v$ , $w$ , ...)	${\frac {(v-2)(2v-1)(uv+1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-z^{3}a^{7}-3za^{7}-2a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+3za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}+a^{3}z^{-1}-z^{3}a-2za-az^{-1}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+a^{8}z^{6}+a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+15a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}+3a^{6}+4a^{5}z^{7}-9a^{5}z^{5}+11a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-4a^{4}z^{4}+2a^{4}+2a^{3}z^{7}-2a^{3}z^{5}-4a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}+2a^{2}z^{6}-4a^{2}z^{4}+a^{2}+az^{5}-3az^{3}+3az-az^{-1}$ (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

1

-1

-2

4

1

3

-4

2

0

-6

4

3

1

-8

3

2

-1

-10

2

4

-2

-12

2

4

2

-14

1

-1

-16

2

Integral Khovanov Homology

(db, data source)

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