L11a328

(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_20,5,21,6 X_18,9,19,10 X_22,19,9,20 X_16,12,17,11 X_6,21,7,22 X_14,8,15,7 X_4,14,5,13 X_8,16,1,15 X_2,17,3,18
Gauss code	{1, -11, 2, -9, 3, -7, 8, -10}, {4, -1, 6, -2, 9, -8, 10, -6, 11, -4, 5, -3, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 4 v 2 u 3 -2 v u 3 + 3 v 3 u 2 -9 v 2 u 2 + 7 v u 2 - u 2 - v 3 u + 7 v 2 u -9 v u + 3 u -2 v 2 + 4 v -2$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+6 q^{7/2}-11 q^{5/2}+15 q^{3/2}-18 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- a z 7 - z 7 a -1 + a 3 z 5 -3 a z 5 -4 z 5 a -1 + z 5 a -3 + 2 a 3 z 3 - a z 3 -6 z 3 a -1 + 3 z 3 a -3 + 2 a z -4 z a -1 + 2 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -5 a 3 z 9 -11 a z 9 -6 z 9 a -1 -4 a 4 z 8 -6 a 2 z 8 -8 z 8 a -2 -10 z 8 - a 5 z 7 + 14 a 3 z 7 + 28 a z 7 + 6 z 7 a -1 -7 z 7 a -3 + 14 a 4 z 6 + 37 a 2 z 6 + 13 z 6 a -2 -5 z 6 a -4 + 41 z 6 + 3 a 5 z 5 -5 a 3 z 5 -9 a z 5 + 10 z 5 a -1 + 8 z 5 a -3 -3 z 5 a -5 -14 a 4 z 4 -38 a 2 z 4 -8 z 4 a -2 + 4 z 4 a -4 - z 4 a -6 -37 z 4 -3 a 5 z 3 -6 a 3 z 3 -11 a z 3 -15 z 3 a -1 -4 z 3 a -3 + 3 z 3 a -5 + 4 a 4 z 2 + 10 a 2 z 2 + z 2 a -2 - z 2 a -4 + z 2 a -6 + 9 z 2 + a 5 z + 3 a 3 z + 6 a z + 6 z a -1 + z a -3 - z a -5 + 1- a z -1 - a -1 z -1$ (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 L11a328. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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