L9a9

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	[edit L9a9 Quick Notes] L9a9 is $9_{37}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 X_4,17,1,18 X_12,6,13,5 X_10,4,11,3 X_18,12,5,11 X_2,14,3,13
Gauss code	{1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}-q^{7/2}+{\frac {5}{q^{7/2}}}+3q^{5/2}-{\frac {7}{q^{5/2}}}-5q^{3/2}+{\frac {8}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+6{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+az^{7}+5az^{5}-z^{5}a^{-1}+8az^{3}-3z^{3}a^{-1}+4az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{8}-2z^{8}-4a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4a^{4}z^{6}-3z^{6}a^{-2}+z^{6}-3a^{5}z^{5}+7a^{3}z^{5}+22az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+5a^{4}z^{4}+3a^{2}z^{4}+7z^{4}a^{-2}+4z^{4}+4a^{5}z^{3}-8a^{3}z^{3}-24az^{3}-10z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+4a^{3}z+8az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

3

1

2

3

2

-1

0

6

3

-2

4

5

1

-4

3

4

-1

-6

2

4

2

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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