L10a90

(Knotscape image)

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

Visit L10a90's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + 2 v 3 u 2 -5 v 2 u 2 + 4 v u 2 - u 2 - v 3 u + 4 v 2 u -5 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+10 q^{3/2}-11 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -5 z 5 a -1 + z 5 a -3 - a 3 z 3 + 7 a z 3 -9 z 3 a -1 + 3 z 3 a -3 -2 a 3 z + 6 a z -6 z a -1 + 2 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -3 z 8 a -2 -6 z 8 -3 a 3 z 7 -5 a z 7 -6 z 7 a -1 -4 z 7 a -3 - a 4 z 6 + 7 a 2 z 6 + z 6 a -2 -4 z 6 a -4 + 13 z 6 + 10 a 3 z 5 + 22 a z 5 + 18 z 5 a -1 + 3 z 5 a -3 -3 z 5 a -5 + 3 a 4 z 4 - a 2 z 4 + 3 z 4 a -2 + 4 z 4 a -4 - z 4 a -6 -6 z 4 -9 a 3 z 3 -22 a z 3 -17 z 3 a -1 + 4 z 3 a -5 -2 a 4 z 2 -2 a 2 z 2 -3 z 2 a -2 - z 2 a -4 + z 2 a -6 - z 2 + 3 a 3 z + 8 a z + 6 z a -1 - z a -5 + 1- a z -1 - a -1 z -1$ (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 =

1 is the signature of L10a90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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