L9n27

(Knotscape image)

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

Visit L9n27's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,15 X₈₄₉₃ X_5,17,6,16 X_17,5,18,14 X_15,10,16,11 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -5, -4, 8, 9, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q 3 - q 2 + q + 1 + q -1 + q -2 + q -4 - q -5$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 2 a 4 -2 a 4 + z 4 a 2 + 5 z 2 a 2 + a 2 z -2 + 6 a 2 - z 4 -5 z 2 -2 z -2 -6 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a z 7 + z 7 a -1 + a 4 z 6 + 2 a 2 z 6 + z 6 a -2 + 2 z 6 + a 5 z 5 + a 3 z 5 -5 a z 5 -5 z 5 a -1 -5 a 4 z 4 -13 a 2 z 4 -5 z 4 a -2 -13 z 4 -4 a 5 z 3 -6 a 3 z 3 + 2 a z 3 + 4 z 3 a -1 + 6 a 4 z 2 + 22 a 2 z 2 + 6 z 2 a -2 + 22 z 2 + 2 a 5 z + 6 a 3 z + 6 a z + 2 z a -1 -4 a 4 -12 a 2 -4 a -2 -11-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

-1 is the signature of L9n27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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