L9n28

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	[edit L9n28 Quick Notes] L9n28 is $9_{20}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,15,9 X₈₄₉₃ X_5,17,6,16 X_17,5,18,14 X_10,16,11,15 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -5, 4, -8, 9, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)^{2}(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$3q^{5}-4q^{4}+6q^{3}-5q^{2}+6q-4+3q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+5z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}+3a^{-2}-4a^{-4}+a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$2z^{7}a^{-1}+2z^{7}a^{-3}+8z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}+az^{5}-z^{5}a^{-1}+z^{5}a^{-3}+3z^{5}a^{-5}-21z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}-2az^{3}-7z^{3}a^{-1}-8z^{3}a^{-3}-3z^{3}a^{-5}+17z^{2}a^{-2}+18z^{2}a^{-4}+6z^{2}a^{-6}+5z^{2}+az+3za^{-1}+7za^{-3}+5za^{-5}-7a^{-2}-10a^{-4}-5a^{-6}-1-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

χ

11

3

9

3

2

-1

7

3

1

2

5

2

3

1

3

4

3

1

2

4

2

-1

1

2

-1

-3

2

-5

1

-1

Integral Khovanov Homology

(db, data source)

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