L11a359

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 -2 v 3 u 3 + 5 v 2 u 3 -4 v u 3 + u 3 - v 4 u 2 + 5 v 3 u 2 -9 v 2 u 2 + 5 v u 2 - u 2 + v 4 u -4 v 3 u + 5 v 2 u -2 v u + v 3 - v 2$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-6 q^{9/2}+10 q^{7/2}-14 q^{5/2}+15 q^{3/2}-16 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 + 2 z 5 a -1 + z 5 a -3 - a 3 z 3 + a z 3 + 4 z 3 a -1 + z 3 a -3 - z 3 a -5 - a 3 z + 3 z a -1 - z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 -3 a z 9 -6 z 9 a -1 -3 z 9 a -3 -4 a 2 z 8 -6 z 8 a -2 -5 z 8 a -4 -5 z 8 -3 a 3 z 7 + 4 a z 7 + 12 z 7 a -1 -5 z 7 a -5 - a 4 z 6 + 10 a 2 z 6 + 15 z 6 a -2 + 9 z 6 a -4 -3 z 6 a -6 + 14 z 6 + 9 a 3 z 5 -14 z 5 a -1 + 7 z 5 a -3 + 11 z 5 a -5 - z 5 a -7 + 3 a 4 z 4 -6 a 2 z 4 -15 z 4 a -2 -7 z 4 a -4 + 6 z 4 a -6 -11 z 4 -7 a 3 z 3 + 3 a z 3 + 12 z 3 a -1 -9 z 3 a -3 -9 z 3 a -5 + 2 z 3 a -7 -2 a 4 z 2 + a 2 z 2 + 5 z 2 a -2 + z 2 a -4 -2 z 2 a -6 + 5 z 2 + a 3 z -2 a z -5 z a -1 + z a -3 + 3 z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 =

1 is the signature of L11a359. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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