L11a382

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 4 u 4 + 2 v 3 u 4 - v 2 u 4 + v 4 u 3 -5 v 3 u 3 + 5 v 2 u 3 - v u 3 + 4 v 3 u 2 -9 v 2 u 2 + 4 v u 2 - v 3 u + 5 v 2 u -5 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -7 a z 7 + z 7 a -1 + 5 a 3 z 5 -18 a z 5 + 5 z 5 a -1 + 8 a 3 z 3 -20 a z 3 + 8 z 3 a -1 + 4 a 3 z -9 a z + 4 z a -1 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -9 a z 9 -5 z 9 a -1 -4 a 4 z 8 -5 z 8 a -2 - z 8 -4 a 5 z 7 + 8 a 3 z 7 + 31 a z 7 + 16 z 7 a -1 -3 z 7 a -3 -3 a 6 z 6 + 3 a 4 z 6 + 6 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 17 z 6 - a 7 z 5 + 5 a 5 z 5 -14 a 3 z 5 -47 a z 5 -18 z 5 a -1 + 9 z 5 a -3 + 6 a 6 z 4 + 3 a 4 z 4 -12 a 2 z 4 -14 z 4 a -2 + 3 z 4 a -4 -26 z 4 + 2 a 7 z 3 + a 5 z 3 + 14 a 3 z 3 + 30 a z 3 + 10 z 3 a -1 -5 z 3 a -3 -3 a 6 z 2 - a 4 z 2 + 6 a 2 z 2 + 5 z 2 a -2 - z 2 a -4 + 10 z 2 - a 7 z - a 5 z -5 a 3 z -9 a z -3 z a -1 + z a -3 - a 2 + a 3 z -1 + a z -1$ (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 L11a382. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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