L10n72

(Knotscape image)

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

Visit L10n72's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_9,11,10,20 X_11,18,12,19 X_15,9,16,8 X_19,5,20,10 X_17,14,18,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, {-5, -2, 10, 8, -6, 3, -8, 5, -7, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 2 + w u 2 + u 2 + v 2 u - v u + v w u - w u - v 2 + v - v 2 w$ (db)
Jones polynomial	$q 6 + 2 q 3 - q 2 + 3 q -2 + 2 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 4 + a 2 z 2 + 2 z 2 a -2 - z 2 a -4 -3 z 2 + a 2 + 4 a -2 -4 a -4 + a -6 -2 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 3 z 7 a -1 + z 7 a -3 + a 2 z 6 -5 z 6 a -2 + z 6 a -6 -3 z 6 -9 a z 5 -15 z 5 a -1 -6 z 5 a -3 -4 a 2 z 4 + 6 z 4 a -2 -2 z 4 a -4 -6 z 4 a -6 -2 z 4 + 9 a z 3 + 19 z 3 a -1 + 8 z 3 a -3 -2 z 3 a -5 + 3 a 2 z 2 - z 2 a -2 + 6 z 2 a -4 + 9 z 2 a -6 + 5 z 2 -2 a z -6 z a -1 + 4 z a -5 - a 2 -3 a -2 -6 a -4 -4 a -6 -1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L10n72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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