L10n40

(Knotscape image)

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

Visit L10n40'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_11,16,12,17 X_6,18,1,17 X_14,19,15,20 X_18,13,19,14
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 $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 + 2 v u 3 - u 3 - v u 2 - v 2 u + 2 v u + v 2 - v$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+2 q^{5/2}-4 q^{3/2}+3 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + 2 a z 3 -4 z 3 a -1 + z 3 a -3 - a 3 z + 5 a z -5 z a -1 + 2 z a -3 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 -2 a z 7 -4 z 7 a -1 -2 z 7 a -3 - a 2 z 6 + 2 z 6 a -2 - z 6 a -4 + 2 z 6 + 9 a z 5 + 18 z 5 a -1 + 9 z 5 a -3 + 3 a 2 z 4 + 4 z 4 a -2 + 4 z 4 a -4 + 3 z 4 -2 a 3 z 3 -16 a z 3 -25 z 3 a -1 -11 z 3 a -3 - a 4 z 2 -6 a 2 z 2 -5 z 2 a -2 -3 z 2 a -4 -7 z 2 + 2 a 3 z + 11 a z + 14 z a -1 + 5 z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 =

-1 is the signature of L10n40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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