L10n86

(Knotscape image)

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

Visit L10n86's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,15 X_16,8,17,7 X_8,16,9,15 X_11,18,12,19 X_19,12,20,13 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}

A Braid Representative

A Morse Link Presentation

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

-4 is the signature of L10n86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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