L10a139

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_14,8,15,7 X_8,14,9,13 X_20,15,13,16 X_16,19,17,20 X_12,18,5,17 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 $u$ , $v$ , $w$ , ...)	$- v u 3 + 2 u 3 + 4 v u 2 -2 v w u 2 + 3 w u 2 -4 u 2 -3 v u + 4 v w u -4 w u + 2 u -2 v w + w$ (db)
Jones polynomial	$q 4 -2 q 3 + 5 q 2 -7 q + 11-10 q -1 + 10 q -2 -8 q -3 + 6 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -3 z 2 a 4 -2 a 4 + 2 z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2 + z 4 -2 z 2 -2 z -2 -3-2 z 2 a -2 + a -2 z -2 + a -4$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 3 a 5 z 7 + 6 a 3 z 7 + 6 a z 7 + 3 z 7 a -1 + a 6 z 6 -5 a 4 z 6 -11 a 2 z 6 + 3 z 6 a -2 -2 z 6 -9 a 5 z 5 -24 a 3 z 5 -17 a z 5 + 2 z 5 a -3 -3 a 6 z 4 -5 a 4 z 4 - a 2 z 4 -3 z 4 a -2 + z 4 a -4 -3 z 4 + 7 a 5 z 3 + 21 a 3 z 3 + 11 a z 3 -5 z 3 a -1 -2 z 3 a -3 + 3 a 6 z 2 + 6 a 4 z 2 + 8 a 2 z 2 + 3 z 2 a -2 -2 z 2 a -4 + 10 z 2 -2 a 5 z -6 a 3 z + 2 a z + 6 z a -1 - a 6 - a 4 -4 a 2 -3 a -2 + a -4 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L10a139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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