L8n6

(Knotscape image)

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

Visit L8n6's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,13 X_7,14,8,15 X_13,8,14,9 X_15,12,16,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 3 - v w u 2 + u + v w$ (db)
Jones polynomial	$q -2 + q -6 + q -7 + q -9$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 10 z -2 -2 a 8 z -2 -2 a 8 + a 6 z -2 + z 4 a 4 + 4 z 2 a 4 + 2 a 4$ (db)
Kauffman polynomial	$z 6 a 10 -6 z 4 a 10 + 10 z 2 a 10 + a 10 z -2 -6 a 10 + z 5 a 9 -6 z 3 a 9 + 8 z a 9 -2 a 9 z -1 + z 6 a 8 -7 z 4 a 8 + 14 z 2 a 8 + 2 a 8 z -2 -9 a 8 + z 5 a 7 -6 z 3 a 7 + 8 z a 7 -2 a 7 z -1 + a 6 z -2 -2 a 6 + z 4 a 4 -4 z 2 a 4 + 2 a 4$ (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 =

-3 is the signature of L8n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$	$i = -1$
$r = -8$		${\mathbb Z}$	${\mathbb Z}$
$r = -7$
$r = -6$		${\mathbb Z}$
$r = -5$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$