L10n8

(Knotscape image)

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

Visit L10n8's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,5,12,20 X_13,19,14,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, 10, -2, -3, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u + 2 u + 2 v -2$ (db)
Jones polynomial	$-q^{9/2}+q^{7/2}-2 q^{5/2}+3 q^{3/2}-3 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 3 + a 3 z -1 - z 3 a -2 z a - a z -1 - z 3 a -1 - z a -1 - a -1 z -1 + 2 z a -3 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 - a z 7 -2 z 7 a -1 - z 7 a -3 - a 2 z 6 + 6 z 6 a -2 + 5 z 6 - a 3 z 5 + 3 a z 5 + 10 z 5 a -1 + 6 z 5 a -3 + 3 a 2 z 4 -13 z 4 a -2 - z 4 a -4 -9 z 4 + 4 a 3 z 3 -17 z 3 a -1 -14 z 3 a -3 - z 3 a -5 + 9 z 2 a -2 + 2 z 2 a -4 + 7 z 2 -3 a 3 z -3 a z + 8 z a -1 + 11 z a -3 + 3 z a -5 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L10n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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