L10n80

(Knotscape image)

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

Visit L10n80's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_7,14,8,15 X_13,8,14,9 X_15,13,16,20 X_19,17,20,16 X_17,12,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8}, {-5, 4, -6, 7, -8, 3, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 -2 u 3 -2 v u 2 - w u 2 + 2 u 2 + v u -2 v w u + 2 w u + 2 v w - w$ (db)
Jones polynomial	$2 q -1 -4 q -2 + 5 q -3 -5 q -4 + 6 q -5 -4 q -6 + 4 q -7 - q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 10 z -2 -2 a 8 z -2 -3 a 8 + 3 z 2 a 6 + a 6 z -2 + 3 a 6 - z 4 a 4 - z 2 a 4 - a 4 + 2 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 6 a 10 -5 z 4 a 10 + 8 z 2 a 10 + a 10 z -2 -5 a 10 + z 7 a 9 -2 z 5 a 9 -3 z 3 a 9 + 6 z a 9 -2 a 9 z -1 + z 8 a 8 - z 6 a 8 -6 z 4 a 8 + 10 z 2 a 8 + 2 a 8 z -2 -7 a 8 + 4 z 7 a 7 -11 z 5 a 7 + 5 z 3 a 7 + 2 z a 7 -2 a 7 z -1 + z 8 a 6 + z 6 a 6 -6 z 4 a 6 + 3 z 2 a 6 + a 6 z -2 -2 a 6 + 3 z 7 a 5 -8 z 5 a 5 + 11 z 3 a 5 -6 z a 5 + 3 z 6 a 4 -5 z 4 a 4 + 4 z 2 a 4 + z 5 a 3 + 3 z 3 a 3 -2 z a 3 + 3 z 2 a 2 - a 2$ (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 =

-2 is the signature of L10n80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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