L10n19

(Knotscape image)

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

Visit L10n19's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_11,14,12,15 X₃₈₄₉ X_5,13,6,12 X_13,5,14,20 X_9,16,10,17 X_15,10,16,11 X_17,2,18,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -8, 9, -4, 6, -7, 4, -9, 8, -10, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 t(1) t(2)^3-4 t(1) t(2)^2+t(2)^2+t(1) t(2)-4 t(2)+2}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$\frac{5}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -1 - a 7 z 3 -3 a 7 z -2 a 7 z -1 + a 5 z 5 + 3 a 5 z 3 + 2 a 5 z + a 5 z -1 + a 3 z 5 + 3 a 3 z 3 + 2 a 3 z + a 3 z -1 - a z 3 -3 a z - a z -1$ (db)
Kauffman polynomial	$3 a 9 z 3 -5 a 9 z + a 9 z -1 + a 8 z 6 -2 a 8 z 2 + a 8 + 2 a 7 z 7 -8 a 7 z 5 + 18 a 7 z 3 -13 a 7 z + 2 a 7 z -1 + a 6 z 8 -3 a 6 z 6 + 8 a 6 z 4 -7 a 6 z 2 + 3 a 6 + 3 a 5 z 7 -10 a 5 z 5 + 16 a 5 z 3 -9 a 5 z + a 5 z -1 + a 4 z 8 -3 a 4 z 6 + 6 a 4 z 4 -6 a 4 z 2 + 2 a 4 + a 3 z 7 - a 3 z 5 -3 a 3 z 3 + 3 a 3 z - a 3 z -1 + a 2 z 6 -2 a 2 z 4 - a 2 z 2 + a 2 + a z 5 -4 a z 3 + 4 a z - a z -1$ (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 =

-3 is the signature of L10n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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