L10n81

(Knotscape image)

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

Visit L10n81's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_3,10,4,11 X_5,14,6,15 X_11,20,12,13 X_13,12,14,5 X_19,1,20,4 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v 3 u 2 -2 v 2 u 2 + v u 2 + 2 v 2 w u 2 -2 v w u 2 + 2 v 2 u -2 v u - v 2 w u + 2 v w u - w u + v$ (db)
Jones polynomial	$2 q -2 -2 q -3 + 6 q -4 -6 q -5 + 6 q -6 -6 q -7 + 5 q -8 -2 q -9 + q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 4 a 8 + 3 z 2 a 8 + a 8 z -2 + 4 a 8 - z 6 a 6 -5 z 4 a 6 -11 z 2 a 6 -2 a 6 z -2 -11 a 6 + 2 z 4 a 4 + 7 z 2 a 4 + a 4 z -2 + 7 a 4$ (db)
Kauffman polynomial	$z 4 a 12 -2 z 2 a 12 + a 12 + 2 z 5 a 11 -2 z 3 a 11 + 3 z 6 a 10 -4 z 4 a 10 + 2 z 2 a 10 + 2 z 7 a 9 -2 z 3 a 9 + z 8 a 8 + 4 z 4 a 8 -7 z 2 a 8 - a 8 z -2 + 5 a 8 + 3 z 7 a 7 -5 z 5 a 7 + 9 z 3 a 7 -11 z a 7 + 2 a 7 z -1 + z 8 a 6 -3 z 6 a 6 + 12 z 4 a 6 -20 z 2 a 6 -2 a 6 z -2 + 13 a 6 + z 7 a 5 -3 z 5 a 5 + 9 z 3 a 5 -11 z a 5 + 2 a 5 z -1 + 3 z 4 a 4 -9 z 2 a 4 - a 4 z -2 + 8 a 4$ (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 =

-4 is the signature of L10n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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