L8a19

(Knotscape image)

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

Visit L8a19's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_8,12,9,11 X_16,8,11,7 X_10,13,5,14 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v 2 u + 3 v u + v 2 w u -3 v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$q 4 -3 q 3 + 5 q 2 -5 q + 8-5 q -1 + 5 q -2 -3 q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + a 2 z 4 + z 4 a -2 -4 z 4 + 2 a 2 z 2 + 2 z 2 a -2 -5 z 2 + a 2 + a -2 -2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 7 + 2 z 7 a -1 + 4 a 2 z 6 + 4 z 6 a -2 + 8 z 6 + 3 a 3 z 5 + a z 5 + z 5 a -1 + 3 z 5 a -3 + a 4 z 4 -7 a 2 z 4 -7 z 4 a -2 + z 4 a -4 -16 z 4 -4 a 3 z 3 -4 a z 3 -4 z 3 a -1 -4 z 3 a -3 - a 4 z 2 + 5 a 2 z 2 + 5 z 2 a -2 - z 2 a -4 + 12 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L8a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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