L10a100

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_14,19,15,20 X_18,7,19,8 X_16,5,17,6 X_4,15,5,16 X_6,17,7,18 X_20,13,9,14 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -9, 2, -6, 5, -7, 4, -10}, {9, -1, 10, -2, 8, -3, 6, -5, 7, -4, 3, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 3 + v u 3 -2 v 3 u 2 + 3 v 2 u 2 -3 v u 2 + u 2 + v 3 u -3 v 2 u + 3 v u -2 u + v 2 -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 11 + 2 z a 11 - z 5 a 9 -2 z 3 a 9 + z a 9 + a 9 z -1 -2 z 5 a 7 -7 z 3 a 7 -6 z a 7 - a 7 z -1 - z 5 a 5 -3 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 5 a 15 + 3 z 3 a 15 -2 z a 15 -2 z 6 a 14 + 5 z 4 a 14 -2 z 2 a 14 -2 z 7 a 13 + 3 z 5 a 13 + z 3 a 13 -2 z 8 a 12 + 5 z 6 a 12 -8 z 4 a 12 + 7 z 2 a 12 - z 9 a 11 + z 7 a 11 - z 5 a 11 - z a 11 -4 z 8 a 10 + 13 z 6 a 10 -19 z 4 a 10 + 7 z 2 a 10 - z 9 a 9 + 6 z 5 a 9 -13 z 3 a 9 + 5 z a 9 - a 9 z -1 -2 z 8 a 8 + 4 z 6 a 8 - z 4 a 8 -3 z 2 a 8 + a 8 -3 z 7 a 7 + 10 z 5 a 7 -12 z 3 a 7 + 7 z a 7 - a 7 z -1 -2 z 6 a 6 + 5 z 4 a 6 - z 2 a 6 - z 5 a 5 + 3 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L10a100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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