L11a324

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 2 v 2 u 3 + 2 v 3 u 2 -6 v 2 u 2 + 3 v u 2 + 3 v 2 u -6 v u + 2 u + 2 v -2$ (db)
Jones polynomial	$q^{7/2}-2 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 - a z 7 + a 5 z 5 -5 a 3 z 5 -5 a z 5 + z 5 a -1 + 4 a 5 z 3 -8 a 3 z 3 -8 a z 3 + 4 z 3 a -1 + 4 a 5 z -5 a 3 z -5 a z + 4 z a -1 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -5 a z 9 -2 z 9 a -1 -5 a 4 z 8 -3 a 2 z 8 - z 8 a -2 + z 8 -5 a 5 z 7 + 5 a 3 z 7 + 21 a z 7 + 11 z 7 a -1 -4 a 6 z 6 + 14 a 4 z 6 + 24 a 2 z 6 + 6 z 6 a -2 + 12 z 6 -3 a 7 z 5 + 11 a 5 z 5 + 11 a 3 z 5 -23 a z 5 -20 z 5 a -1 -2 a 8 z 4 + 5 a 6 z 4 -10 a 4 z 4 -29 a 2 z 4 -12 z 4 a -2 -24 z 4 - a 9 z 3 + 2 a 7 z 3 -8 a 5 z 3 -18 a 3 z 3 + 8 a z 3 + 15 z 3 a -1 + a 8 z 2 -2 a 6 z 2 + 3 a 4 z 2 + 9 a 2 z 2 + 8 z 2 a -2 + 11 z 2 + a 9 z - a 7 z + 5 a 5 z + 8 a 3 z -4 a z -5 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-3 is the signature of L11a324. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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