L10n49

(Knotscape image)

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

Visit L10n49's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_11,16,12,17 X_14,20,15,19 X_17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, 7, 8, -10, -2, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v 2 u 3 + v u 2 + 2 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - z 5 a 7 -5 z 3 a 7 -6 z a 7 -2 a 7 z -1 + z 7 a 5 + 6 z 5 a 5 + 11 z 3 a 5 + 8 z a 5 + 2 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -6 z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$- z a 11 - z 2 a 10 -2 z 3 a 9 + 4 z a 9 - a 9 z -1 - z 6 a 8 + 3 z 4 a 8 - z 2 a 8 -2 z 7 a 7 + 10 z 5 a 7 -15 z 3 a 7 + 10 z a 7 -2 a 7 z -1 - z 8 a 6 + 4 z 6 a 6 -3 z 4 a 6 + z 2 a 6 - a 6 -3 z 7 a 5 + 16 z 5 a 5 -24 z 3 a 5 + 12 z a 5 -2 a 5 z -1 - z 8 a 4 + 5 z 6 a 4 -6 z 4 a 4 + z 2 a 4 - z 7 a 3 + 6 z 5 a 3 -11 z 3 a 3 + 7 z a 3 - a 3 z -1$ (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 =

-5 is the signature of L10n49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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