L8a2

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	[edit L8a2 Quick Notes] L8a2 is $8_{10}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-5q^{5/2}+5q^{3/2}-6{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}-2az^{3}+3z^{3}a^{-1}-z^{3}a^{-3}+a^{3}z-4az+4za^{-1}-za^{-3}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+3z^{4}a^{-4}-z^{2}a^{-4}+a^{3}z^{5}+5z^{5}a^{-3}-3a^{3}z^{3}-6z^{3}a^{-3}+3a^{3}z+3za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{6}+4z^{6}a^{-2}-5a^{2}z^{4}-4z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}+az^{7}+z^{7}a^{-1}+3az^{5}+7z^{5}a^{-1}-13az^{3}-17z^{3}a^{-1}+10az+10za^{-1}-2az^{-1}-2a^{-1}z^{-1}+6z^{6}-12z^{4}+5z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

3

1

2

4

2

0

2

4

3

1

0

3

4

1

-2

1

2

-1

-4

1

3

2

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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