L10n32

(Knotscape image)

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

Visit L10n32's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{\sqrt{q}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 - z 3 a 3 -3 z a 3 -2 a 3 z -1 + a z -1 + z 3 a -1 + 3 z a -1 + a -1 z -1 - z a -3 - a -3 z -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -2 a 3 z 7 - a z 7 + 6 a 4 z 6 + 7 a 2 z 6 - z 6 a -2 + 6 a 5 z 5 + 13 a 3 z 5 + 7 a z 5 - z 5 a -1 - z 5 a -3 -10 a 4 z 4 -14 a 2 z 4 + 4 z 4 a -2 -10 a 5 z 3 -23 a 3 z 3 -12 a z 3 + 5 z 3 a -1 + 4 z 3 a -3 + 6 a 4 z 2 + 11 a 2 z 2 -2 z 2 a -2 + 3 z 2 + 5 a 5 z + 13 a 3 z + 6 a z -5 z a -1 -3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 =

0 is the signature of L10n32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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