L9a36

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	[edit L9a36 Quick Notes] L9a36 is $9_{1}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{2}+uv-u-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+2q^{7/2}-2q^{5/2}+q^{3/2}+q^{19/2}-2q^{17/2}+2q^{15/2}-3q^{13/2}+3q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-7}+4z^{3}a^{-7}+3za^{-7}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-7za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+5z^{6}a^{-4}+3z^{6}a^{-6}-2z^{6}a^{-8}+6z^{5}a^{-3}+15z^{5}a^{-5}+7z^{5}a^{-7}-2z^{5}a^{-9}-6z^{4}a^{-4}+4z^{4}a^{-8}-2z^{4}a^{-10}-11z^{3}a^{-3}-21z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-2z^{3}a^{-11}-2z^{2}a^{-6}+z^{2}a^{-10}-z^{2}a^{-12}+7za^{-3}+9za^{-5}+2za^{-7}+za^{-9}+za^{-11}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

20

1

-1

18

1

16

1

0

14

2

1

12

2

0

10

1

0

8

1

2

1

6

1

0

4

1

2

1

2

0

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=4$	$i=6$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$