L8a5

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	[edit L8a5 Quick Notes] L8a5 is $8_{11}^{2}$ 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_14,8,15,7 X_16,10,5,9 X_8,16,9,15 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -5, 4, -6, 8, -2, 6, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

2

1

4

3

2

-1

2

0

3

4

1

-2

1

0

-4

3

-6

1

0

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=2}
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$