L8n5

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	[edit L8n5 Quick Notes] L8n5 is is $8_{9}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,16,10,11 X_11,10,12,5 X_4,15,1,16
Gauss code	{1, -4, 3, -8}, {-2, -1, 5, -3, -6, 7}, {-7, 2, 4, -5, 8, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

2

0

-5

2

-7

2

-9

3

2

1

-11

1

2

1

-13

1

-1

-15

1

Integral Khovanov Homology

(db, data source)

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