L8a8

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	[edit L8a8 Quick Notes] L8a8 is $8_{7}^{2}$ in the Rolfsen table of links, and the "seized Carrick bend" of practical knot-tying.

[edit L8a8 Further Notes and Views]

The simplest Celtic or pseudo-Celtic linear decorative knot.		Alternate decorative variant		Circular arcs only		Decorative variant with big loops at ends
Coat of arms of Bressauc, Jura, Switzerland.

Link Presentations

[edit Notes on L8a8's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_16,10,7,9 X₂₇₃₈ X_14,12,15,11 X_12,5,13,6 X_4,13,5,14 X_6,16,1,15
Gauss code	{1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

3

2

1

0

2

4

2

-2

2

0

-4

1

3

2

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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