L8a17

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	[edit L8a17 Quick Notes] L8a17 is $8_{2}^{3}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

2

1

-1

-7

2

-9

2

0

-11

4

2

-13

1

3

2

-15

3

0

-17

1

3

2

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

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