L10n57

(Knotscape image)

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

Visit L10n57's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_20,5,9,6 X_3,15,4,14 X_15,5,16,4 X_7,17,8,16 X_11,18,12,19 X_17,12,18,13 X_2,9,3,10 X_13,1,14,8 X_6,19,7,20
Gauss code	{1, -8, -3, 4, 2, -10, -5, 9}, {8, -1, -6, 7, -9, 3, -4, 5, -7, 6, 10, -2}

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	$-q^{7/2}+q^{5/2}-\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a z 5 - a 3 z 3 + 5 a z 3 -3 a 3 z + 5 a z - z a -1 - z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 7 - a z 7 - a 4 z 6 - a 2 z 6 - z 6 a -2 - z 6 + 6 a 3 z 5 + 7 a z 5 - z 5 a -3 + 5 a 4 z 4 + 6 a 2 z 4 + 5 z 4 a -2 + 6 z 4 -10 a 3 z 3 -14 a z 3 + 4 z 3 a -3 -5 a 4 z 2 -7 a 2 z 2 -5 z 2 a -2 -7 z 2 + 6 a 3 z + 10 a z + 2 z a -1 -2 z a -3 + 1- a z -1 - a -1 z -1$ (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 =

0 is the signature of L10n57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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