L8n2

(Knotscape image)

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

Visit L8n2's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_9,12,10,13 X₃₈₄₉ X_5,11,6,10 X_11,5,12,16 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

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 3 + a 3 z -1 - z 3 a -3 z a -2 a z -1 + 2 z a -1 + 2 a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- a 2 z 6 - z 6 - a 3 z 5 -2 a z 5 - z 5 a -1 + 4 a 2 z 4 + 4 z 4 + 4 a 3 z 3 + 8 a z 3 + 4 z 3 a -1 -3 a 2 z 2 - z 2 a -2 -4 z 2 -3 a 3 z -8 a z -6 z a -1 - z a -3 + 1 + a 3 z -1 + 2 a z -1 + 2 a -1 z -1 + a -3 z -1$ (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 L8n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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