L8a9

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	[edit L8a9 Quick Notes] L8a9 is $8_{8}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a9's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_16,10,7,9 X₂₇₃₈ X_4,16,5,15 X_12,5,13,6 X_14,11,15,12 X_6,13,1,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

1

-1

0

4

2

-2

3

0

-4

3

0

-6

2

4

2

-8

1

2

-1

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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