L10n84

(Knotscape image)

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

Visit L10n84's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,15 X_7,16,8,17 X_15,8,16,9 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 $u$ , $v$ , $w$ , ...)	$- v u 4 + v u 3 - v 2 w u 3 - u 3 + v 2 w u - v w u + u + v w$ (db)
Jones polynomial	$q -3 + q -6 - q -7 + 2 q -8 - q -9 + 2 q -10 - q -11 + q -12$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 12 z -2 + a 12 -2 z 2 a 10 -2 a 10 z -2 -4 a 10 - z 2 a 8 + a 8 z -2 + z 6 a 6 + 6 z 4 a 6 + 9 z 2 a 6 + 3 a 6$ (db)
Kauffman polynomial	$z 6 a 14 -5 z 4 a 14 + 6 z 2 a 14 -2 a 14 + z 7 a 13 -4 z 5 a 13 + 2 z 3 a 13 + z a 13 + z 8 a 12 -5 z 6 a 12 + 7 z 4 a 12 -6 z 2 a 12 - a 12 z -2 + 4 a 12 + 2 z 7 a 11 -11 z 5 a 11 + 17 z 3 a 11 -10 z a 11 + 2 a 11 z -1 + z 8 a 10 -7 z 6 a 10 + 17 z 4 a 10 -20 z 2 a 10 -2 a 10 z -2 + 10 a 10 + z 7 a 9 -7 z 5 a 9 + 14 z 3 a 9 -10 z a 9 + 2 a 9 z -1 - z 4 a 8 + z 2 a 8 - a 8 z -2 + 2 a 8 - z 3 a 7 + z a 7 + z 6 a 6 -6 z 4 a 6 + 9 z 2 a 6 -3 a 6$ (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 L10n84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$	$i = -3$
$r = -10$		${\mathbb Z}$	${\mathbb Z}$
$r = -9$		${\mathbb Z}$
$r = -8$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}^{3}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$