L11a288

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_16,9,17,10 X_20,12,21,11 X_22,15,9,16 X_14,6,15,5 X_4,14,5,13 X_6,20,7,19 X_18,8,19,7 X_8,18,1,17 X_2,21,3,22
Gauss code	{1, -11, 2, -7, 6, -8, 9, -10}, {3, -1, 4, -2, 7, -6, 5, -3, 10, -9, 8, -4, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 3 v 2 u 4 -3 v u 4 + u 4 + 2 v 3 u 3 -5 v 2 u 3 + 6 v u 3 -2 u 3 -2 v 3 u 2 + 6 v 2 u 2 -5 v u 2 + 2 u 2 + v 3 u -3 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+6 q^{11/2}-11 q^{9/2}+13 q^{7/2}-16 q^{5/2}+16 q^{3/2}-14 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -4 z 5 a -1 -4 z 5 a -3 + z 5 a -5 + 3 a z 3 -5 z 3 a -1 -5 z 3 a -3 + 3 z 3 a -5 + 2 a z - z a -1 - z a -3 + 2 z a -5 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -5 z 9 a -1 -9 z 9 a -3 -4 z 9 a -5 -2 z 8 a -2 -4 z 8 a -6 -6 z 8 -5 a z 7 + 12 z 7 a -1 + 29 z 7 a -3 + 9 z 7 a -5 -3 z 7 a -7 -3 a 2 z 6 + 16 z 6 a -2 + 8 z 6 a -4 + 9 z 6 a -6 - z 6 a -8 + 15 z 6 - a 3 z 5 + 11 a z 5 -13 z 5 a -1 -40 z 5 a -3 -6 z 5 a -5 + 9 z 5 a -7 + 6 a 2 z 4 -26 z 4 a -2 -9 z 4 a -4 -2 z 4 a -6 + 3 z 4 a -8 -16 z 4 + 2 a 3 z 3 -7 a z 3 + 2 z 3 a -1 + 21 z 3 a -3 + 3 z 3 a -5 -7 z 3 a -7 - a 2 z 2 + 9 z 2 a -2 + 3 z 2 a -4 - z 2 a -6 -2 z 2 a -8 + 6 z 2 + 2 a z - z a -1 -2 z a -3 + 3 z a -5 + 2 z a -7 + a -4 - a -3 z -1 - a -5 z -1$ (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 L11a288. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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