L9a49

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	[edit L9a49 Quick Notes] L9a49 is $9_{6}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,10,17,9 X_14,8,15,7 X_18,14,11,13 X_10,16,5,15 X_8,18,9,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -7, 3, -6}, {9, -2, 5, -4, 6, -3, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w-uv^{2}+uvw^{2}-3uvw+2uv-uw^{2}+2uw-2v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+q^{-4}-4q^{3}-2q^{-3}+7q^{2}+5q^{-2}-7q-6q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+a^{4}+z^{4}a^{-2}-2a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{4}-z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+3a^{2}z^{6}+4z^{6}a^{-2}+3z^{6}a^{-4}+4z^{6}+2a^{3}z^{5}+2az^{5}-7z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+a^{4}z^{4}-4a^{2}z^{4}-13z^{4}a^{-2}-8z^{4}a^{-4}-10z^{4}-2a^{3}z^{3}-7az^{3}-z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}+4a^{2}z^{2}+11z^{2}a^{-2}+5z^{2}a^{-4}+12z^{2}+6az+6za^{-1}+a^{4}-3a^{2}-5a^{-2}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

2

1

-1

5

2

3

4

0

1

4

3

1

-1

3

5

2

-3

2

3

-1

-5

3

-7

1

2

-1

-9

1

Integral Khovanov Homology

(db, data source)

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