L11a377

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 4 u 4 + 2 v 3 u 4 - v 2 u 4 + v 4 u 3 -5 v 3 u 3 + 5 v 2 u 3 - v u 3 + 4 v 3 u 2 -7 v 2 u 2 + 4 v u 2 - v 3 u + 5 v 2 u -5 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+6 q^{13/2}-10 q^{11/2}+13 q^{9/2}-15 q^{7/2}+14 q^{5/2}-13 q^{3/2}+9 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 9 a -3 + z 7 a -1 -7 z 7 a -3 + z 7 a -5 + 5 z 5 a -1 -18 z 5 a -3 + 5 z 5 a -5 + 8 z 3 a -1 -20 z 3 a -3 + 8 z 3 a -5 + 5 z a -1 -8 z a -3 + 4 z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -4 z 9 a -1 -9 z 9 a -3 -5 z 9 a -5 + z 8 a -2 -2 z 8 a -4 -6 z 8 a -6 -3 z 8 - a z 7 + 15 z 7 a -1 + 32 z 7 a -3 + 10 z 7 a -5 -6 z 7 a -7 + 15 z 6 a -2 + 16 z 6 a -4 + 8 z 6 a -6 -5 z 6 a -8 + 12 z 6 + 4 a z 5 -16 z 5 a -1 -40 z 5 a -3 -11 z 5 a -5 + 6 z 5 a -7 -3 z 5 a -9 -20 z 4 a -2 -16 z 4 a -4 -3 z 4 a -6 + 5 z 4 a -8 - z 4 a -10 -13 z 4 -4 a z 3 + 8 z 3 a -1 + 27 z 3 a -3 + 11 z 3 a -5 - z 3 a -7 + 3 z 3 a -9 + 7 z 2 a -2 + 6 z 2 a -4 -2 z 2 a -8 + z 2 a -10 + 4 z 2 + a z -4 z a -1 -9 z a -3 -4 z a -5 - z a -7 - z a -9 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11a377. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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