L9a40

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	[edit L9a40 Quick Notes] L9a40 is $9_{4}^{2}$ in the Rolfsen table of links.

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National swiss museum

Povray depiction

Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_8,9,1,10 X_18,13,9,14 X_14,8,15,7 X_16,6,17,5 X_6,16,7,15 X_4,18,5,17
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -6, 8, -7, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v+uv^{2}+uv+u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-2za^{5}+z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{5}a+4z^{3}a+4za+az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+2a^{7}z^{3}+3a^{6}z^{4}-2a^{6}z^{2}+4a^{5}z^{5}-8a^{5}z^{3}+4a^{5}z+3a^{4}z^{6}-6a^{4}z^{4}+a^{4}z^{2}+2a^{3}z^{7}-4a^{3}z^{5}-2a^{3}z^{3}+2a^{3}z+a^{2}z^{8}-2a^{2}z^{6}-2a^{2}z^{4}+2a^{2}z^{2}+3az^{7}+z^{7}a^{-1}-14az^{5}-6z^{5}a^{-1}+19az^{3}+11z^{3}a^{-1}-8az+az^{-1}-6za^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+7z^{4}-2z^{2}-1$ (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

1

-1

-2

3

2

1

-4

2

3

1

-6

2

1

-8

1

2

1

-10

1

2

-1

-12

1

-14

1

-1

Integral Khovanov Homology

(db, data source)

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