L9a38

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	[edit L9a38 Quick Notes] L9a38 is $9_{5}^{2}$ 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_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_14,6,15,5 X_18,14,9,13 X_16,8,17,7 X_6,16,7,15 X_8,18,1,17
Gauss code	{1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+3q^{7/2}-{\frac {1}{q^{7/2}}}-5q^{5/2}+{\frac {2}{q^{5/2}}}+5q^{3/2}-{\frac {3}{q^{3/2}}}+q^{11/2}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-6z^{5}a^{-1}+z^{5}a^{-3}+4az^{3}-12z^{3}a^{-1}+4z^{3}a^{-3}+4az-8za^{-1}+4za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-2z^{2}a^{-6}+2z^{5}a^{-5}-4z^{3}a^{-5}+za^{-5}+2z^{6}a^{-4}-3z^{4}a^{-4}+z^{2}a^{-4}+2z^{7}a^{-3}+a^{3}z^{5}-5z^{5}a^{-3}-3a^{3}z^{3}+8z^{3}a^{-3}+a^{3}z-4za^{-3}+z^{8}a^{-2}+2a^{2}z^{6}-z^{6}a^{-2}-6a^{2}z^{4}+3a^{2}z^{2}+3z^{2}a^{-2}+2az^{7}+4z^{7}a^{-1}-6az^{5}-14z^{5}a^{-1}+6az^{3}+21z^{3}a^{-1}-4az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-z^{6}-2z^{4}+3z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

1

8

2

1

-1

6

3

1

2

4

2

0

2

4

3

1

0

2

4

2

-2

1

2

-1

-4

1

2

1

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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