L9a19

(Knotscape image)

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

Visit L9a19's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_10,5,11,6 X_18,11,5,12 X_4,17,1,18 X_16,14,17,13 X_14,8,15,7 X_8,16,9,15
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -6 u 2 -6 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 + a 5 z -1 - z 5 a 3 -2 z 3 a 3 -3 z a 3 - a 3 z -1 - z 5 a - z 3 a + z 3 a -1$ (db)
Kauffman polynomial	$-2 a 4 z 8 -2 a 2 z 8 -4 a 5 z 7 -10 a 3 z 7 -6 a z 7 -3 a 6 z 6 -4 a 4 z 6 -8 a 2 z 6 -7 z 6 - a 7 z 5 + 7 a 5 z 5 + 18 a 3 z 5 + 6 a z 5 -4 z 5 a -1 + 6 a 6 z 4 + 13 a 4 z 4 + 17 a 2 z 4 - z 4 a -2 + 9 z 4 + 2 a 7 z 3 -4 a 5 z 3 -10 a 3 z 3 - a z 3 + 3 z 3 a -1 -3 a 6 z 2 -7 a 4 z 2 -6 a 2 z 2 -2 z 2 - a 7 z + 2 a 5 z + 3 a 3 z + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L9a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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