L8n6

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	[edit L8n6 Quick Notes] L8n6 is $8_{10}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,13 X_7,14,8,15 X_13,8,14,9 X_15,12,16,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

1

-7

1

0

-9

0

-11

1

3

1

-13

2

-15

1

-17

1

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$	$i=-1$
$r=-8$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$
$r=-6$		${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$