L8a15

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	[edit L8a15 Quick Notes] L8a15 is $8_{3}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a15's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw-2uv-2uw+2u-2vw+2v+2w-1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{-7}-q^{-6}+4q^{-5}-4q^{-4}+6q^{-3}-4q^{-2}+q+4q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+3a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{4}-a^{2}z^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-3za^{7}+2a^{7}z^{-1}+z^{6}a^{6}+2z^{4}a^{6}-6z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{7}a^{5}+2z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+4z^{6}a^{4}-5z^{4}a^{4}-a^{4}z^{-2}+3a^{4}+z^{7}a^{3}+2z^{5}a^{3}-3z^{3}a^{3}+3z^{6}a^{2}-5z^{4}a^{2}+2z^{2}a^{2}+3z^{5}a-5z^{3}a+z^{4}-z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

2

1

-3

3

0

-5

3

1

2

-7

1

3

2

-9

3

0

-11

1

4

3

-13

0

-15

1

Integral Khovanov Homology

(db, data source)

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