L9n19

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	[edit L9n19 Quick Notes] L9n19 is $9_{61}^{2}$ 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_3,12,4,13 X_18,5,9,6 X_6,9,7,10 X_16,12,17,11 X_7,14,8,15 X_13,4,14,5 X_15,8,16,1 X_2,17,3,18
Gauss code	{1, -9, -2, 7, 3, -4, -6, 8}, {4, -1, 5, 2, -7, 6, -8, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(1)^{2}+t(2)\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-a^{7}z^{-1}-z^{5}a^{5}-5z^{3}a^{5}-5za^{5}$ (db)
Kauffman polynomial	$a^{11}z^{5}-5a^{11}z^{3}+5a^{11}z-a^{9}z+a^{9}z^{-1}-a^{8}-a^{7}z+a^{7}z^{-1}+a^{5}z^{5}-5a^{5}z^{3}+5a^{5}z$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

-8

1

0

-10

0

-12

1

2

1

0

-14

0

-16

1

0

-18

1

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$	$i=-2$
$r=-8$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$
$r=-6$		${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$