L8n4

(Knotscape image)

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

Visit L8n4's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,16,10,11 X_11,10,12,5 X_15,1,16,4
Gauss code	{1, -4, -3, 8}, {-2, -1, 5, 3, -6, 7}, {-7, 2, 4, -5, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 2 + v 2 u - v u + v w u - w u + v$ (db)
Jones polynomial	$2 q -1 - q -2 + 3 q -3 -2 q -4 + 2 q -5 - q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 6 + a 6 z -2 + 2 a 6 - z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -6 a 4 + 2 z 2 a 2 + a 2 z -2 + 4 a 2$ (db)
Kauffman polynomial	$z 4 a 8 -3 z 2 a 8 + a 8 + z 5 a 7 -2 z 3 a 7 + z 6 a 6 -3 z 4 a 6 + 4 z 2 a 6 + a 6 z -2 -3 a 6 + 2 z 5 a 5 -5 z 3 a 5 + 6 z a 5 -2 a 5 z -1 + z 6 a 4 -4 z 4 a 4 + 10 z 2 a 4 + 2 a 4 z -2 -8 a 4 + z 5 a 3 -3 z 3 a 3 + 6 z a 3 -2 a 3 z -1 + 3 z 2 a 2 + a 2 z -2 -5 a 2$ (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 =

-2 is the signature of L8n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -6$	${\mathbb Z}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{2}$	${\mathbb Z}^{2}$