L10n87

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 - v w u 3 + v 2 u 2 - v u 2 + v w u 2 - w u 2 + v u + w$ (db)
Jones polynomial	$q 7 - q 6 + 2 q 5 -2 q 4 + 2 q 3 - q 2 + q + 1 + q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 6 a -2 -7 z 4 a -2 + z 4 -14 z 2 a -2 + 2 z 2 a -4 + z 2 a -6 + 5 z 2 -10 a -2 + 3 a -4 + a -6 + 6-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + z 7 a -1 + z 7 a -3 -9 z 6 a -2 + z 6 a -6 -8 z 6 -8 z 5 a -1 -8 z 5 a -3 + z 5 a -5 + z 5 a -7 + 26 z 4 a -2 + z 4 a -4 -3 z 4 a -6 + z 4 a -8 + 21 z 4 + 18 z 3 a -1 + 19 z 3 a -3 - z 3 a -5 -2 z 3 a -7 -30 z 2 a -2 -2 z 2 a -4 + 3 z 2 a -6 -3 z 2 a -8 -22 z 2 -13 z a -1 -13 z a -3 + 14 a -2 + 4 a -4 - a -6 + a -8 + 9 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L10n87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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