L10n56

(Knotscape image)

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

Visit L10n56's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_18,11,19,12 X_20,5,9,6 X_7,15,8,14 X_12,4,13,3 X_13,16,14,17 X_15,7,16,6 X_8,9,1,10 X_4,19,5,20 X_2,18,3,17
Gauss code	{1, -10, 5, -9, 3, 7, -4, -8}, {8, -1, 2, -5, -6, 4, -7, 6, 10, -2, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 - v u 2 + u 2 + v u - u + v -1$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z a 5 - z a 3 + z 5 a + 5 z 3 a + 5 z a + a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 - a 3 z 7 -2 a z 7 - z 7 a -1 + 6 a 2 z 6 + 6 z 6 - a 5 z 5 + 6 a 3 z 5 + 13 a z 5 + 6 z 5 a -1 - a 6 z 4 -8 a 2 z 4 -9 z 4 + 4 a 5 z 3 -8 a 3 z 3 -22 a z 3 -10 z 3 a -1 + 3 a 6 z 2 + a 4 z 2 + a 2 z 2 + 3 z 2 -2 a 5 z + 2 a 3 z + 10 a z + 6 z a -1 + 1- a z -1 - a -1 z -1$ (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 =

-2 is the signature of L10n56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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