L8a11

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	[edit L8a11 Quick Notes] L8a11 is $8_{3}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_16,11,7,12 X_12,15,13,16 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+2za^{9}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+2z^{2}a^{12}-2z^{5}a^{11}+z^{3}a^{11}-2z^{6}a^{10}+3z^{4}a^{10}-3z^{2}a^{10}-z^{7}a^{9}+z^{5}a^{9}-2z^{3}a^{9}+2za^{9}-3z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}-z^{7}a^{7}+2z^{5}a^{7}-za^{7}+a^{7}z^{-1}-z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

2

-10

1

0

-12

3

2

1

-14

1

0

-16

2

3

-1

-18

1

2

1

-20

1

-1

-22

1

Integral Khovanov Homology

(db, data source)

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