L10n54

(Knotscape image)

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

Visit L10n54's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_17,5,18,4 X_5,19,6,18 X_6,14,7,13 X_14,8,15,7 X_8,16,1,15 X_19,17,20,16
Gauss code	{1, -4, 2, 5, -6, -7, 8, -9}, {4, -1, 3, -2, 7, -8, 9, 10, -5, 6, -10, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 + v 3 u 2 + u + v -1$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+q^{11/2}-2 q^{9/2}+2 q^{7/2}-2 q^{5/2}+q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 + z 5 a -3 -6 z 5 a -5 + z 5 a -7 + 5 z 3 a -3 -11 z 3 a -5 + 5 z 3 a -7 + 6 z a -3 -8 z a -5 + 5 z a -7 - z a -9 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - z 7 a -3 -3 z 7 a -5 -2 z 7 a -7 + 5 z 6 a -4 + 4 z 6 a -6 - z 6 a -8 + 6 z 5 a -3 + 16 z 5 a -5 + 10 z 5 a -7 -6 z 4 a -4 -2 z 4 a -6 + 4 z 4 a -8 -11 z 3 a -3 -23 z 3 a -5 -12 z 3 a -7 + z 2 a -4 - z 2 a -6 -2 z 2 a -8 + 7 z a -3 + 10 z a -5 + 3 z a -7 + a -4 - a -3 z -1 - a -5 z -1$ (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 =

5 is the signature of L10n54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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