L9n16

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	[edit L9n16 Quick Notes] L9n16 is $9_{51}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,17,12,16 X_3,10,4,11 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_14,10,15,9 X_18,14,7,13 X_17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -7, 4, 2, -9, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

1

-2

2

0

-4

1

-6

1

2

1

-8

1

0

-10

1

-12

1

0

-14

1

Integral Khovanov Homology

(db, data source)

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