L10n85

(Knotscape image)

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

Visit L10n85's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_20,14,15,13 X_7,16,8,17 X_15,8,16,9 X_18,12,19,11 X_12,20,13,19 X_14,18,5,17 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 $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 -2 v 2 u + 3 v u + v 2 w u -3 v w u + 2 w u - u - v + v w - w + 1$ (db)
Jones polynomial	$q 4 -3 q 3 + 5 q 2 -6 q + 8-6 q -1 + 6 q -2 -3 q -3 + 2 q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + a 2 z 4 + z 4 a -2 -4 z 4 + a 2 z 2 + 2 z 2 a -2 -5 z 2 + a 4 -2 a 2 + a -2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 2 z 8 + z 8 + a 3 z 7 + 4 a z 7 + 3 z 7 a -1 - a 2 z 6 + 4 z 6 a -2 + 3 z 6 - a 3 z 5 -7 a z 5 -3 z 5 a -1 + 3 z 5 a -3 + 3 a 4 z 4 + 4 a 2 z 4 -6 z 4 a -2 + z 4 a -4 -6 z 4 + 2 a 3 z 3 + 7 a z 3 + z 3 a -1 -4 z 3 a -3 -7 a 4 z 2 -8 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 3 z 2 -5 a 3 z -5 a z + 4 a 4 + 6 a 2 - a -2 + 2 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L10n85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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