L10a152

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 + 2 v 2 u 3 -3 v u 3 - v 2 w u 3 + 2 v w u 3 + u 3 -2 v 2 u 2 + 4 v u 2 + 2 v 2 w u 2 -4 v w u 2 + 2 w u 2 -2 u 2 -2 v u - v 2 w u + 3 v w u -2 w u + u - v w$ (db)
Jones polynomial	$q 7 -3 q 6 + 7 q 5 -9 q 4 + 12 q 3 -12 q 2 + 11 q -8 + 6 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 2 z 4 a -2 -2 z 4 a -4 -2 z 4 + a 2 z 2 - z 2 a -2 -3 z 2 a -4 + z 2 a -6 -4 z 2 + 2 a 2 -4 a -2 + a -4 + a -6 -2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 9 a -1 + z 9 a -3 + 7 z 8 a -2 + 4 z 8 a -4 + 3 z 8 + 2 a z 7 + 6 z 7 a -1 + 10 z 7 a -3 + 6 z 7 a -5 + a 2 z 6 -15 z 6 a -2 + 6 z 6 a -6 -8 z 6 -5 a z 5 -23 z 5 a -1 -27 z 5 a -3 -6 z 5 a -5 + 3 z 5 a -7 -4 a 2 z 4 + 15 z 4 a -2 -7 z 4 a -4 -9 z 4 a -6 + z 4 a -8 + 8 z 4 + 2 a z 3 + 23 z 3 a -1 + 24 z 3 a -3 + z 3 a -5 -2 z 3 a -7 + 5 a 2 z 2 -18 z 2 a -2 + 3 z 2 a -4 + 8 z 2 a -6 - z 2 a -8 -7 z 2 + a z -10 z a -1 -10 z a -3 + z a -5 -2 a 2 + 10 a -2 + 2 a -4 -3 a -6 + 4 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L10a152. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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