L11a381

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 4 -2 v 2 u 4 + v u 4 -5 v 3 u 3 + 9 v 2 u 3 -5 v u 3 + u 3 - v 4 u 2 + 8 v 3 u 2 -15 v 2 u 2 + 8 v u 2 - u 2 + v 4 u -5 v 3 u + 9 v 2 u -5 v u + v 3 -2 v 2 + v$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-10 q^{9/2}+17 q^{7/2}-23 q^{5/2}+26 q^{3/2}-27 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -2 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 3 a z 3 + 3 z 3 a -3 - z 3 a -5 - a 3 z + 3 z a -1 - z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 -8 a z 9 -17 z 9 a -1 -9 z 9 a -3 -8 a 2 z 8 -18 z 8 a -2 -12 z 8 a -4 -14 z 8 -4 a 3 z 7 + 11 a z 7 + 27 z 7 a -1 + 3 z 7 a -3 -9 z 7 a -5 - a 4 z 6 + 17 a 2 z 6 + 46 z 6 a -2 + 18 z 6 a -4 -4 z 6 a -6 + 42 z 6 + 8 a 3 z 5 -7 z 5 a -1 + 15 z 5 a -3 + 13 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -12 a 2 z 4 -33 z 4 a -2 -12 z 4 a -4 + 4 z 4 a -6 -31 z 4 -5 a 3 z 3 + 2 z 3 a -1 -13 z 3 a -3 -9 z 3 a -5 + z 3 a -7 - a 4 z 2 + 4 a 2 z 2 + 9 z 2 a -2 + 4 z 2 a -4 - z 2 a -6 + 9 z 2 + a 3 z -2 a z -5 z a -1 + z a -3 + 3 z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11a381. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$