L9a37

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	[edit L9a37 Quick Notes] L9a37 is $9_{7}^{2}$ 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_2,9,3,10 X_4,18,5,17 X_16,8,17,7 X_14,6,15,5 X_6,16,7,15 X_8,14,1,13
Gauss code	{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

18

1

-1

16

2

14

3

1

-2

12

4

2

10

4

0

8

4

3

1

6

2

4

2

4

2

4

-2

2

1

3

2

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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