L11a321

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 4 v 2 u 3 -2 v u 3 + 3 v 3 u 2 -11 v 2 u 2 + 9 v u 2 - u 2 - v 3 u + 9 v 2 u -11 v u + 3 u -2 v 2 + 4 v -2$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-7 q^{3/2}+12 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 3 a 3 z 5 + 4 a z 5 - z 5 a -1 -2 a 5 z 3 + 2 a 3 z 3 + 7 a z 3 -3 z 3 a -1 - a 5 z - a 3 z + 5 a z -3 z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -6 a 5 z 9 -11 a 3 z 9 -5 a z 9 -7 a 6 z 8 -10 a 4 z 8 -9 a 2 z 8 -6 z 8 -4 a 7 z 7 + 8 a 5 z 7 + 19 a 3 z 7 + 2 a z 7 -5 z 7 a -1 - a 8 z 6 + 16 a 6 z 6 + 32 a 4 z 6 + 25 a 2 z 6 -3 z 6 a -2 + 7 z 6 + 9 a 7 z 5 + 5 a 5 z 5 -4 a 3 z 5 + 8 a z 5 + 7 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -9 a 6 z 4 -24 a 4 z 4 -21 a 2 z 4 + 5 z 4 a -2 -3 z 4 -5 a 7 z 3 -5 a 5 z 3 -9 a 3 z 3 -15 a z 3 -4 z 3 a -1 + 2 z 3 a -3 - a 8 z 2 + 3 a 6 z 2 + 5 a 4 z 2 + 3 a 2 z 2 -2 z 2 a -2 + a 7 z + a 5 z + 4 a 3 z + 8 a z + 3 z a -1 - z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$