L10n68

(Knotscape image)

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

Visit L10n68's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_9,18,10,19 X_11,16,12,17 X_17,20,18,11 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 t(1) t(3)^2 t(2)^2-t(1) t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)+t(3)-2}{\sqrt{t(1)} t(2) t(3)}$ (db)
Jones polynomial	$q -3 - q -4 + 3 q -5 -2 q -6 + 4 q -7 -3 q -8 + 3 q -9 -2 q -10 + q -11$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a 12 - a 10 z 4 -4 a 10 z 2 + a 10 z -2 -2 a 10 + a 8 z 6 + 4 a 8 z 4 + 2 a 8 z 2 -2 a 8 z -2 -3 a 8 + a 6 z 6 + 5 a 6 z 4 + 7 a 6 z 2 + a 6 z -2 + 4 a 6$ (db)
Kauffman polynomial	$a 14 z 2 - a 14 + 2 a 13 z 3 -2 a 13 z + a 12 z 6 -3 a 12 z 4 + 5 a 12 z 2 - a 12 + 2 a 11 z 7 -9 a 11 z 5 + 14 a 11 z 3 -6 a 11 z + a 10 z 8 -3 a 10 z 6 + a 10 z 4 + 3 a 10 z 2 + a 10 z -2 -3 a 10 + 3 a 9 z 7 -12 a 9 z 5 + 11 a 9 z 3 -2 a 9 z -1 + a 8 z 8 -3 a 8 z 6 - a 8 z 4 + 6 a 8 z 2 + 2 a 8 z -2 -6 a 8 + a 7 z 7 -3 a 7 z 5 - a 7 z 3 + 4 a 7 z -2 a 7 z -1 + a 6 z 6 -5 a 6 z 4 + 7 a 6 z 2 + a 6 z -2 -4 a 6$ (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 =

-6 is the signature of L10n68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$	$i = -3$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$