L8a6

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	[edit L8a6 Quick Notes] L8a6 is $8_{6}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a6's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,5,7 X_14,10,15,9 X_10,14,11,13 X_8,16,9,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

1

-1

4

1

0

2

1

-1

0

2

1

-2

2

3

1

-4

1

0

-6

2

-8

1

0

-10

1

Integral Khovanov Homology

(db, data source)

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