L8n8

(Knotscape image)

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

Visit L8n8's page at the original Knot Atlas.

Detail from an 18th century royal decree, Vietnam.

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_16,11,13,12 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_12,15,9,16
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u w + v w + u x - v x$ (db)
Jones polynomial	$-q^{7/2}-q^{3/2}-2 \sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z a 3 + 2 a 3 z -1 + a 3 z -3 - z 3 a -5 z a -6 a z -1 -3 a z -3 + z 3 a -1 + 5 z a -1 + 6 a -1 z -1 + 3 a -1 z -3 - z a -3 -2 a -3 z -1 - a -3 z -3$ (db)
Kauffman polynomial	$- a 3 z 5 - a z 5 - z 5 a -1 - z 5 a -3 - a 2 z 4 - z 4 a -2 -2 z 4 + 5 a 3 z 3 + 7 a z 3 + 7 z 3 a -1 + 5 z 3 a -3 + 6 a 2 z 2 + 6 z 2 a -2 + 12 z 2 -6 a 3 z -14 a z -14 z a -1 -6 z a -3 -8 a 2 -8 a -2 -15 + 4 a 3 z -1 + 9 a z -1 + 9 a -1 z -1 + 4 a -3 z -1 + 3 a 2 z -2 + 3 a -2 z -2 + 6 z -2 - a 3 z -3 -3 a z -3 -3 a -1 z -3 - a -3 z -3$ (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 =

0 is the signature of L8n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$	$i = 2$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$
$r = -2$		${\mathbb Z}$
$r = -1$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{3}$	${\mathbb Z}^{6}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 3$
$r = 4$	${\mathbb Z}$	${\mathbb Z}$