L8a1

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	[edit L8a1 Quick Notes] L8a1 is $8_{13}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a1's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+5{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)+2a^{3}z^{3}+2a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z+4a^{4}z^{6}-5a^{4}z^{4}+2a^{4}z^{2}+2a^{3}z^{7}+4a^{3}z^{5}-10a^{3}z^{3}+4a^{3}z+9a^{2}z^{6}-14a^{2}z^{4}+z^{4}a^{-2}+5a^{2}z^{2}+2az^{7}+5az^{5}+4z^{5}a^{-1}-11az^{3}-5z^{3}a^{-1}+2az+az^{-1}+a^{-1}z^{-1}+5z^{6}-7z^{4}+2z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

1

-1

0

5

3

2

-2

4

0

-4

3

0

-6

2

4

2

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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