L8n5

(Knotscape image)

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

Visit L8n5's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,16,10,11 X_11,10,12,5 X_4,15,1,16
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 $u$ , $v$ , $w$ , ...)	$v u - v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$2 q -1 -2 q -2 + 4 q -3 -2 q -4 + 3 q -5 -2 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 6 + a 6 z -2 + a 6 - z 4 a 4 -3 z 2 a 4 -2 a 4 z -2 -4 a 4 + 2 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z 4 a 8 -2 z 2 a 8 + a 8 + 2 z 5 a 7 -4 z 3 a 7 + z 6 a 6 -2 z 2 a 6 + a 6 z -2 - a 6 + 3 z 5 a 5 -5 z 3 a 5 + 4 z a 5 -2 a 5 z -1 + z 6 a 4 - z 4 a 4 + 3 z 2 a 4 + 2 a 4 z -2 -4 a 4 + z 5 a 3 - z 3 a 3 + 4 z a 3 -2 a 3 z -1 + 3 z 2 a 2 + a 2 z -2 -3 a 2$ (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 L8n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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