L10a137

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_18,10,19,9 X_4,17,1,18 X_12,20,5,19 X_20,12,13,11 X_10,14,11,13
Gauss code	{1, -4, 3, -7}, {2, -1, 5, -3, 6, -10, 9, -8}, {10, -2, 4, -5, 7, -6, 8, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 + v 3 u 2 -3 v 2 u 2 + 3 v u 2 - v 3 w u 2 + 3 v 2 w u 2 -3 v w u 2 + w u 2 - u 2 - v 3 u + 3 v 2 u -3 v u + v 3 w u -3 v 2 w u + 3 v w u - w u + u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -5 q 6 + 10 q 5 -12 q 4 + 14 q 3 -12 q 2 + 11 q -7 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 3 z 4 a -4 - z 4 a -6 - z 4 + 3 z 2 a -4 -2 z 2 a -6 - z 2 - a -4 + 1 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 9 a -3 + 2 z 9 a -5 + 5 z 8 a -2 + 9 z 8 a -4 + 4 z 8 a -6 + 6 z 7 a -1 + 4 z 7 a -3 + 2 z 7 a -5 + 4 z 7 a -7 -6 z 6 a -2 -19 z 6 a -4 -6 z 6 a -6 + 3 z 6 a -8 + 4 z 6 + a z 5 -10 z 5 a -1 -9 z 5 a -3 -6 z 5 a -5 -7 z 5 a -7 + z 5 a -9 + z 4 a -2 + 18 z 4 a -4 + 3 z 4 a -6 -7 z 4 a -8 -7 z 4 - a z 3 + 2 z 3 a -1 + z 3 a -3 + 3 z 3 a -5 + 3 z 3 a -7 -2 z 3 a -9 -2 z 2 a -2 -5 z 2 a -4 + 3 z 2 a -6 + 4 z 2 a -8 + 2 z 2 + 4 z a -3 + 4 z a -5 - a -2 -4 a -4 -3 a -6 + 1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L10a137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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