L9n6

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	[edit L9n6 Quick Notes] L9n6 is $9_{55}^{2}$ 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_4,15,1,16 X_5,12,6,13 X₈₄₉₃ X_9,16,10,17 X_11,18,12,5 X_17,10,18,11 X_2,14,3,13
Gauss code	{1, -9, 5, -3}, {-4, -1, 2, -5, -6, 8, -7, 4, 9, -2, 3, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

0

-6

3

-8

1

2

1

-10

3

0

-12

1

0

-14

1

3

-2

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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