L8a8

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a8's page at the original Knot Atlas.

The simplest Celtic or pseudo-Celtic linear decorative knot.

Alternate decorative variant

Decorative variant with big loops at ends

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

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

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

1 is the signature of L8a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$