L9a28

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	[edit L9a28 Quick Notes] L9a28 is $9_{20}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_18,16,7,15 X_16,12,17,11 X_12,18,13,17 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}+q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-12a^{3}z-3a^{3}z^{-1}+az^{5}+4az^{3}+4az+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z+2a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+3a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-11a^{4}z^{4}+10a^{4}z^{2}-3a^{4}+5a^{3}z^{7}-18a^{3}z^{5}+22a^{3}z^{3}-14a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-10a^{2}z^{4}+12a^{2}z^{2}-3a^{2}+2az^{7}-8az^{5}+9az^{3}-4az+az^{-1}+z^{6}-4z^{4}+4z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

2

1

-1

-2

3

1

2

-4

3

0

-6

3

2

1

-8

2

3

1

-10

2

3

-1

-12

2

-14

1

2

-1

-16

1

Integral Khovanov Homology

(db, data source)

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