L8a16

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	[edit L8a16 Quick Notes] L8a16 is $8_{5}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a16's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_16,10,11,9 X_10,12,5,11 X_4,15,1,16
Gauss code	{1, -4, 3, -8}, {2, -1, 5, -3, 6, -7}, {7, -2, 4, -5, 8, -6}

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

Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

2

1

-1

7

4

2

5

3

4

1

3

2

1

2

5

3

-1

1

0

-3

2

-5

1

-1

Integral Khovanov Homology

(db, data source)

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