L9a39

(Knotscape image)

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

Visit L9a39's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_14,5,15,6 X_6,9,7,10 X_16,8,17,7 X_18,16,9,15 X_8,18,1,17 X_4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 + v 3 u 2 -3 v 2 u 2 + v u 2 + v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$q^{3/2}-2 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 + a 5 z 5 -6 a 3 z 5 + a z 5 + 4 a 5 z 3 -12 a 3 z 3 + 4 a z 3 + 4 a 5 z -9 a 3 z + 3 a z + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$- z 3 a 9 + z a 9 -2 z 4 a 8 + 2 z 2 a 8 -2 z 5 a 7 + z 3 a 7 -2 z 6 a 6 + 2 z 4 a 6 - z 2 a 6 -2 z 7 a 5 + 5 z 5 a 5 -7 z 3 a 5 + 5 z a 5 - a 5 z -1 - z 8 a 4 + z 6 a 4 + 3 z 4 a 4 -4 z 2 a 4 + a 4 -4 z 7 a 3 + 16 z 5 a 3 -20 z 3 a 3 + 10 z a 3 - a 3 z -1 - z 8 a 2 + 2 z 6 a 2 + 3 z 4 a 2 -4 z 2 a 2 -2 z 7 a + 9 z 5 a -11 z 3 a + 4 z a - z 6 + 4 z 4 -3 z 2$ (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 =

-3 is the signature of L9a39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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