L8a12

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	[edit L8a12 Quick Notes] L8a12 is $8_{2}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_16,7,9,8 X_12,3,13,4 X_14,5,15,6 X_4,13,5,14 X_6,15,7,16 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

1

-10

1

0

-12

2

1

-14

1

0

-16

1

2

-1

-18

1

-20

1

0

-22

1

Integral Khovanov Homology

(db, data source)

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