L10n18

(Knotscape image)

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

Visit L10n18's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,12,6,13 X₈₄₉₃ X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_11,20,12,5 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -7, 8, -9, 4, -6, 7, -8, 6, 10, -2, 3, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+2 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 5 - z 5 a -1 - a 3 z 3 + 6 a z 3 -5 z 3 a -1 + z 3 a -3 -3 a 3 z + 9 a z -8 z a -1 + 2 z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 - a 3 z 7 -2 a z 7 -3 z 7 a -1 -2 z 7 a -3 - a 4 z 6 - a 2 z 6 + 3 z 6 a -2 - z 6 a -4 + 4 z 6 + 6 a 3 z 5 + 14 a z 5 + 17 z 5 a -1 + 9 z 5 a -3 + 5 a 4 z 4 + 8 a 2 z 4 + 4 z 4 a -2 + 4 z 4 a -4 + 3 z 4 -9 a 3 z 3 -26 a z 3 -26 z 3 a -1 -9 z 3 a -3 -5 a 4 z 2 -12 a 2 z 2 -10 z 2 a -2 -3 z 2 a -4 -14 z 2 + 6 a 3 z + 17 a z + 15 z a -1 + 4 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 =

-1 is the signature of L10n18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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