L9n18

(Knotscape image)

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

Visit L9n18'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_18,5,9,6 X_7,14,8,15 X_13,16,14,17 X_15,8,16,1 X_6,9,7,10 X_4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, -5, 7}, {8, -1, 2, -3, -6, 5, -7, 6, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 3 v 3 - u v 2 - u 2 v -1$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- z a 11 + z 5 a 9 + 6 z 3 a 9 + 8 z a 9 + a 9 z -1 - z 7 a 7 -7 z 5 a 7 -15 z 3 a 7 -11 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 + 3 z a 13 + z a 11 - z 6 a 10 + 6 z 4 a 10 -8 z 2 a 10 - z 7 a 9 + 7 z 5 a 9 -14 z 3 a 9 + 9 z a 9 - a 9 z -1 - z 6 a 8 + 6 z 4 a 8 -8 z 2 a 8 + a 8 - z 7 a 7 + 7 z 5 a 7 -15 z 3 a 7 + 11 z a 7 - a 7 z -1$ (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 =

-6 is the signature of L9n18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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