L10n64

(Knotscape image)

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

Visit L10n64's page at the original Knot Atlas.

Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X₃₉₄₈ X_17,2,18,3 X_14,6,15,5 X_6,12,7,11 X_9,18,10,19 X_20,15,11,16 X_10,13,1,14 X_4,19,5,20
Gauss code	{1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 -2 v u 3 + u 3 - v 2 u 2 + 3 v u 2 - u 2 - v 3 u + 3 v 2 u - v u + v 3 -2 v 2 + v$ (db)
Jones polynomial	$\sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 2 z 3 a 5 + 3 z a 5 + a 5 z -1 - z 5 a 3 -3 z 3 a 3 -4 z a 3 - a 3 z -1 + z 3 a$ (db)
Kauffman polynomial	$- z 6 a 8 + 3 z 4 a 8 -2 z 2 a 8 -3 z 7 a 7 + 11 z 5 a 7 -10 z 3 a 7 + 2 z a 7 -2 z 8 a 6 + 4 z 6 a 6 + 3 z 4 a 6 -3 z 2 a 6 -7 z 7 a 5 + 24 z 5 a 5 -23 z 3 a 5 + 9 z a 5 - a 5 z -1 -2 z 8 a 4 + 3 z 6 a 4 + 2 z 4 a 4 -2 z 2 a 4 + a 4 -4 z 7 a 3 + 13 z 5 a 3 -17 z 3 a 3 + 8 z a 3 - a 3 z -1 -2 z 6 a 2 + 2 z 4 a 2 -2 z 2 a 2 -4 z 3 a + z a - z 2$ (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 L10n64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}_2$	${\mathbb Z}$