L11a327

(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_18,10,19,9 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_22,18,9,17 X_16,22,17,21 X_4,20,5,19 X_20,6,21,5
Gauss code	{1, -2, 3, -10, 11, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -9, 8, -4, 10, -11, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + v 2 u 5 + v 3 u 4 -4 v 2 u 4 + 2 v u 4 - v 3 u 3 + 5 v 2 u 3 -5 v u 3 + u 3 + v 3 u 2 -5 v 2 u 2 + 5 v u 2 - u 2 + 2 v 2 u -4 v u + u + v -1$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+5 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -8 a z 7 + z 7 a -1 + 6 a 3 z 5 -25 a z 5 + 6 z 5 a -1 + 13 a 3 z 3 -36 a z 3 + 13 z 3 a -1 + 10 a 3 z -22 a z + 10 z a -1 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -6 a z 9 -3 z 9 a -1 -4 a 4 z 8 -4 a 2 z 8 -3 z 8 a -2 -3 z 8 -3 a 5 z 7 + 6 a 3 z 7 + 21 a z 7 + 10 z 7 a -1 -2 z 7 a -3 -2 a 6 z 6 + 10 a 4 z 6 + 17 a 2 z 6 + 9 z 6 a -2 - z 6 a -4 + 15 z 6 - a 7 z 5 + 5 a 5 z 5 -10 a 3 z 5 -43 a z 5 -21 z 5 a -1 + 6 z 5 a -3 + 4 a 6 z 4 -14 a 4 z 4 -30 a 2 z 4 -10 z 4 a -2 + 4 z 4 a -4 -26 z 4 + 3 a 7 z 3 -2 a 5 z 3 + 10 a 3 z 3 + 47 a z 3 + 28 z 3 a -1 -4 z 3 a -3 - a 6 z 2 + 7 a 4 z 2 + 18 a 2 z 2 + 7 z 2 a -2 -4 z 2 a -4 + 21 z 2 -2 a 7 z + a 5 z -9 a 3 z -24 a z -12 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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