L9a34

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	[edit L9a34 Quick Notes] L9a34 is $9_{6}^{2}$ 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_14,5,15,6 X_12,3,13,4 X_16,8,17,7 X_18,15,9,16 X_4,13,5,14 X_6,18,7,17 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -8, 3, -6, 2, -7, 4, -9}, {8, -1, 9, -3, 6, -2, 5, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)+t(2)-1)(t(1)t(2)+1)(t(2)t(1)-t(1)-t(2))}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {6}{q^{11/2}}}-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+z^{5}a^{5}+3z^{3}a^{5}+3za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-za^{3}-a^{3}z^{-1}-z^{3}a-2za$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-2z^{5}a^{9}+4z^{3}a^{9}-2za^{9}-2z^{6}a^{8}+2z^{4}a^{8}-2z^{7}a^{7}+3z^{5}a^{7}-3z^{3}a^{7}-za^{7}-z^{8}a^{6}-z^{2}a^{6}-4z^{7}a^{5}+10z^{5}a^{5}-12z^{3}a^{5}+6za^{5}-a^{5}z^{-1}-z^{8}a^{4}+2z^{4}a^{4}-z^{2}a^{4}+a^{4}-2z^{7}a^{3}+4z^{5}a^{3}-2z^{3}a^{3}+3za^{3}-a^{3}z^{-1}-2z^{6}a^{2}+5z^{4}a^{2}-2z^{2}a^{2}-z^{5}a+3z^{3}a-2za$ (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

1

2

χ

2

1

0

1

-1

-2

3

1

2

-4

3

2

-1

-6

3

2

1

-8

3

0

-10

3

0

-12

1

4

3

-14

1

2

-1

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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