L10n42

(Knotscape image)

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

Visit L10n42's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_16,12,17,11 X_6,18,1,17 X_19,15,20,14 X_13,19,14,18
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 10, -9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + v 2 u 3 - v u 2 + u + v -1$ (db)
Jones polynomial	$q^{15/2}-q^{13/2}+q^{11/2}-2 q^{9/2}+q^{7/2}-2 q^{5/2}+q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 + z 5 a -3 -6 z 5 a -5 + z 5 a -7 + 5 z 3 a -3 -11 z 3 a -5 + 5 z 3 a -7 + 6 z a -3 -9 z a -5 + 5 z a -7 - z a -9 + 2 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - z 7 a -3 -3 z 7 a -5 -2 z 7 a -7 + 5 z 6 a -4 + 4 z 6 a -6 - z 6 a -8 + 6 z 5 a -3 + 17 z 5 a -5 + 11 z 5 a -7 -5 z 4 a -4 + 5 z 4 a -8 -11 z 3 a -3 -27 z 3 a -5 -16 z 3 a -7 -2 z 2 a -4 -7 z 2 a -6 -5 z 2 a -8 + 8 z a -3 + 15 z a -5 + 8 z a -7 + z a -9 + 3 a -4 + 3 a -6 + a -8 -2 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 =

5 is the signature of L10n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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