L9a18

(Knotscape image)

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

Visit L9a18's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,8,5,7 X_16,10,17,9 X_14,12,15,11 X_10,16,11,15 X_8,18,9,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}

A Braid Representative

A Morse Link Presentation

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

1 is the signature of L9a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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