L10a135

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_16,12,17,11 X_10,13,5,14 X_20,18,11,17 X_8,20,9,19 X_18,8,19,7 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, {4, -2, 5, -3, 10, -4, 6, -8, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -2 v u 2 -2 v 2 w u 2 + 3 v w u 2 - w u 2 -3 v 2 u + 6 v u + 2 v 2 w u -6 v w u + 3 w u -2 u + v 2 -3 v + 2 v w -2 w + 2$ (db)
Jones polynomial	$q 6 -4 q 5 + 7 q 4 -10 q 3 + 14 q 2 -13 q + 14-10 q -1 + 7 q -2 -3 q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 + a 2 z 4 -2 z 4 a -2 + z 4 a -4 -3 z 4 + 2 a 2 z 2 + z 2 a -2 + z 2 a -4 -5 z 2 + 2 a 2 + 4 a -2 - a -4 -5 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 z 9 a -1 + 2 z 9 a -3 + 10 z 8 a -2 + 5 z 8 a -4 + 5 z 8 + 7 a z 7 + 8 z 7 a -1 + 5 z 7 a -3 + 4 z 7 a -5 + 6 a 2 z 6 -18 z 6 a -2 -12 z 6 a -4 + z 6 a -6 + z 6 + 3 a 3 z 5 -8 a z 5 -20 z 5 a -1 -20 z 5 a -3 -11 z 5 a -5 + a 4 z 4 -8 a 2 z 4 + 6 z 4 a -4 -2 z 4 a -6 -17 z 4 -2 a 3 z 3 + 3 a z 3 + 6 z 3 a -1 + 8 z 3 a -3 + 7 z 3 a -5 - a 4 z 2 + 8 a 2 z 2 + 11 z 2 a -2 + z 2 a -4 + z 2 a -6 + 20 z 2 + 3 a z + 5 z a -1 + 3 z a -3 + z a -5 -4 a 2 -8 a -2 -2 a -4 -9-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L10a135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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