L11a358

(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_18,8,19,7 X_22,15,11,16 X_6,18,7,17 X_8,20,9,19 X_4,22,5,21 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 2, -9, 4, -7, 5, -8, 3, -11}, {10, -1, 11, -2, 6, -4, 7, -5, 8, -3, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + 3 v 3 u 3 -3 v 2 u 3 + v u 3 + v 4 u 2 -3 v 3 u 2 + 3 v 2 u 2 -3 v u 2 + u 2 + v 3 u -3 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -1 + z 7 a -3 - a z 5 + 5 z 5 a -1 + 4 z 5 a -3 - z 5 a -5 -4 a z 3 + 8 z 3 a -1 + 3 z 3 a -3 -3 z 3 a -5 -3 a z + 6 z a -1 - z a -3 - z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 -2 a z 9 -5 z 9 a -1 -3 z 9 a -3 - a 2 z 8 -2 z 8 a -2 -5 z 8 a -4 + 2 z 8 + 12 a z 7 + 24 z 7 a -1 + 6 z 7 a -3 -6 z 7 a -5 + 6 a 2 z 6 + 19 z 6 a -2 + 11 z 6 a -4 -6 z 6 a -6 + 8 z 6 -24 a z 5 -37 z 5 a -1 + 2 z 5 a -3 + 10 z 5 a -5 -5 z 5 a -7 -11 a 2 z 4 -23 z 4 a -2 -4 z 4 a -4 + 7 z 4 a -6 -3 z 4 a -8 -20 z 4 + 19 a z 3 + 25 z 3 a -1 - z 3 a -3 -2 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 + 6 a 2 z 2 + 7 z 2 a -2 + z 2 a -4 - z 2 a -6 + z 2 a -8 + 10 z 2 -5 a z -9 z a -1 -3 z a -3 - z a -7 - a -2 + a -1 z -1 + a -3 z -1$ (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 =

3 is the signature of L11a358. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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