L10a140

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_10,4,11,3 X_14,6,15,5 X_20,12,13,11 X_12,14,5,13 X_4,19,1,20 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 $u$ , $v$ , $w$ , ...)	$v 3 u 3 - v 2 u 3 - v 3 w u 3 + v 2 w u 3 - v 3 u 2 + 3 v 2 u 2 -2 v u 2 + v 3 w u 2 -3 v 2 w u 2 + 2 v w u 2 -2 v 2 u + 3 v u + 2 v 2 w u -3 v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$- q 5 + 3 q 4 -5 q 3 + 8 q 2 -9 q + 12-9 q -1 + 8 q -2 -5 q -3 + 3 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 - z 6 a -2 + 6 z 6 -4 a 2 z 4 -4 z 4 a -2 + 12 z 4 -4 a 2 z 2 -4 z 2 a -2 + 8 z 2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 4 a 2 z 8 + 4 z 8 a -2 + 8 z 8 + 4 a 3 z 7 -2 a z 7 -2 z 7 a -1 + 4 z 7 a -3 + 3 a 4 z 6 -11 a 2 z 6 -11 z 6 a -2 + 3 z 6 a -4 -28 z 6 + a 5 z 5 -9 a 3 z 5 -2 a z 5 -2 z 5 a -1 -9 z 5 a -3 + z 5 a -5 -7 a 4 z 4 + 14 a 2 z 4 + 14 z 4 a -2 -7 z 4 a -4 + 42 z 4 -2 a 5 z 3 + 4 a 3 z 3 + 6 a z 3 + 6 z 3 a -1 + 4 z 3 a -3 -2 z 3 a -5 + 2 a 4 z 2 -8 a 2 z 2 -8 z 2 a -2 + 2 z 2 a -4 -20 z 2 + 1-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L10a140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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