L10n88

(Knotscape image)

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

Visit L10n88's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_15,17,16,20 X_11,19,12,18 X_17,13,18,12 X_19,5,20,16 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {-6, 5, -7, 4}, {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - w u 3 - u 3 -2 v u 2 + u 2 - v w u + 2 w u + v + v w - w$ (db)
Jones polynomial	$q 3 + 3-2 q -1 + 3 q -2 -3 q -3 + 3 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -2 z 2 a 4 -2 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 4 a 2 - z 4 -6 z 2 -2 z -2 -6 + z 2 a -2 + a -2 z -2 + 3 a -2$ (db)
Kauffman polynomial	$a 4 z 8 + a 2 z 8 + 2 a 5 z 7 + 3 a 3 z 7 + a z 7 + a 6 z 6 -2 a 4 z 6 -4 a 2 z 6 + z 6 a -2 -8 a 5 z 5 -13 a 3 z 5 -5 a z 5 -4 a 6 z 4 -4 a 4 z 4 + 3 a 2 z 4 -6 z 4 a -2 -3 z 4 + 7 a 5 z 3 + 15 a 3 z 3 + 5 a z 3 -3 z 3 a -1 + 4 a 6 z 2 + 4 a 4 z 2 + a 2 z 2 + 9 z 2 a -2 + 10 z 2 -2 a 5 z -6 a 3 z + 2 a z + 6 z a -1 - a 6 -2 a 2 -5 a -2 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L10n88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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