L10a142

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_12,15,5,16 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10 X_20,11,13,12 X_10,19,11,20 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -4, 6, -8, 7, -3}, {10, -2, 3, -5, 4, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 - v 2 w u 3 + v 3 u 2 -2 v 2 u 2 + v u 2 - v 3 w u 2 + 2 v 2 w u 2 - v w u 2 + v 2 u -2 v u - v 2 w u + 2 v w u - w u + u + v - v w + w$ (db)
Jones polynomial	$q -3 -2 q -4 + 4 q -5 -4 q -6 + 7 q -7 -7 q -8 + 7 q -9 -5 q -10 + 4 q -11 -2 q -12 + q -13$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$z 2 a 12 + a 12 z -2 + 3 a 12 -3 z 4 a 10 -12 z 2 a 10 -2 a 10 z -2 -11 a 10 + 2 z 6 a 8 + 10 z 4 a 8 + 15 z 2 a 8 + a 8 z -2 + 8 a 8 + z 6 a 6 + 4 z 4 a 6 + 3 z 2 a 6$ (db)
Kauffman polynomial	$z 4 a 16 -2 z 2 a 16 + a 16 + 2 z 5 a 15 -3 z 3 a 15 + 2 z 6 a 14 - z 4 a 14 -2 z 2 a 14 + 2 z 7 a 13 -2 z 5 a 13 + z 3 a 13 + 2 z 8 a 12 -5 z 6 a 12 + 9 z 4 a 12 -8 z 2 a 12 - a 12 z -2 + 5 a 12 + z 9 a 11 -8 z 5 a 11 + 18 z 3 a 11 -11 z a 11 + 2 a 11 z -1 + 5 z 8 a 10 -22 z 6 a 10 + 38 z 4 a 10 -31 z 2 a 10 -2 a 10 z -2 + 13 a 10 + z 9 a 9 -11 z 5 a 9 + 18 z 3 a 9 -11 z a 9 + 2 a 9 z -1 + 3 z 8 a 8 -14 z 6 a 8 + 23 z 4 a 8 -20 z 2 a 8 - a 8 z -2 + 8 a 8 + 2 z 7 a 7 -7 z 5 a 7 + 4 z 3 a 7 + z 6 a 6 -4 z 4 a 6 + 3 z 2 a 6$ (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 =

-6 is the signature of L10a142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$
$r = -10$	${\mathbb Z}$
$r = -9$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$
$r = -7$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$