L9n19

(Knotscape image)

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

Visit L9n19's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_3,12,4,13 X_18,5,9,6 X_6,9,7,10 X_16,12,17,11 X_7,14,8,15 X_13,4,14,5 X_15,8,16,1 X_2,17,3,18
Gauss code	{1, -9, -2, 7, 3, -4, -6, 8}, {4, -1, 5, 2, -7, 6, -8, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - u 2 - v 3 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - a 7 z -1 - z 5 a 5 -5 z 3 a 5 -5 z a 5$ (db)
Kauffman polynomial	$- z 5 a 11 + 5 z 3 a 11 -5 z a 11 + z a 9 - a 9 z -1 + a 8 + z a 7 - a 7 z -1 - z 5 a 5 + 5 z 3 a 5 -5 z a 5$ (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 =

-4 is the signature of L9n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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