L10a136

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -3 v u 2 - v 2 w u 2 + 3 v w u 2 -2 w u 2 + 2 u 2 -3 v 2 u + 7 v u + 3 v 2 w u -7 v w u + 3 w u -3 u + 2 v 2 -3 v -2 v 2 w + 3 v w - w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -8 q 3 + 13 q 2 -15 q + 18-15 q -1 + 13 q -2 -8 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + 2 a 2 z 4 + 2 z 4 a -2 -2 z 4 - a 4 z 2 + 2 a 2 z 2 + 2 z 2 a -2 - z 2 a -4 -3 z 2 + a 2 + a -2 -2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 6 a 2 z 8 + 6 z 8 a -2 + 12 z 8 + 7 a 3 z 7 + 13 a z 7 + 13 z 7 a -1 + 7 z 7 a -3 + 4 a 4 z 6 -3 a 2 z 6 -3 z 6 a -2 + 4 z 6 a -4 -14 z 6 + a 5 z 5 -11 a 3 z 5 -28 a z 5 -28 z 5 a -1 -11 z 5 a -3 + z 5 a -5 -6 a 4 z 4 -7 a 2 z 4 -7 z 4 a -2 -6 z 4 a -4 -2 z 4 - a 5 z 3 + 5 a 3 z 3 + 12 a z 3 + 12 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + 3 a 4 z 2 + 5 a 2 z 2 + 5 z 2 a -2 + 3 z 2 a -4 + 4 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L10a136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$