L9a40

(Knotscape image)

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

Visit L9a40'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_8,9,1,10 X_18,13,9,14 X_14,8,15,7 X_16,6,17,5 X_6,16,7,15 X_4,18,5,17
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -6, 8, -7, 9, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 - v 3 u 2 + v 2 u 2 - v u 2 + u 2 + v 3 u - v 2 u + v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 5 -2 z a 5 + z 5 a 3 + 3 z 3 a 3 + z a 3 + z 5 a + 4 z 3 a + 4 z a + a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 2 a 8 -2 z 3 a 7 -3 z 4 a 6 + 2 z 2 a 6 -4 z 5 a 5 + 8 z 3 a 5 -4 z a 5 -3 z 6 a 4 + 6 z 4 a 4 - z 2 a 4 -2 z 7 a 3 + 4 z 5 a 3 + 2 z 3 a 3 -2 z a 3 - z 8 a 2 + 2 z 6 a 2 + 2 z 4 a 2 -2 z 2 a 2 -3 z 7 a + 14 z 5 a -19 z 3 a + 8 z a - a z -1 - z 8 + 5 z 6 -7 z 4 + 2 z 2 + 1- z 7 a -1 + 6 z 5 a -1 -11 z 3 a -1 + 6 z a -1 - a -1 z -1$ (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 =

-3 is the signature of L9a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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