L11a340

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(u-1) (v-1) \left(2 u^2 v^2-2 u v^2+7 u v-2 u+2\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-12 q^{9/2}+16 q^{7/2}-\frac{1}{q^{7/2}}-20 q^{5/2}+\frac{3}{q^{5/2}}+19 q^{3/2}-\frac{8}{q^{3/2}}+q^{15/2}-3 q^{13/2}+7 q^{11/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -4 z 5 a -1 -4 z 5 a -3 + z 5 a -5 + 3 a z 3 -7 z 3 a -1 -6 z 3 a -3 + 3 z 3 a -5 + 4 a z -5 z a -1 -2 z a -3 + 3 z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -5 z 9 a -1 -10 z 9 a -3 -5 z 9 a -5 -7 z 8 a -2 -5 z 8 a -4 -5 z 8 a -6 -7 z 8 -6 a z 7 + z 7 a -1 + 21 z 7 a -3 + 11 z 7 a -5 -3 z 7 a -7 -3 a 2 z 6 + 19 z 6 a -2 + 20 z 6 a -4 + 13 z 6 a -6 - z 6 a -8 + 10 z 6 - a 3 z 5 + 10 a z 5 + 12 z 5 a -1 -17 z 5 a -3 -10 z 5 a -5 + 8 z 5 a -7 + 4 a 2 z 4 -16 z 4 a -2 -20 z 4 a -4 -11 z 4 a -6 + 3 z 4 a -8 -6 z 4 + 2 a 3 z 3 -8 a z 3 -17 z 3 a -1 + 6 z 3 a -3 + 8 z 3 a -5 -5 z 3 a -7 - a 2 z 2 + 4 z 2 a -2 + 7 z 2 a -4 + 5 z 2 a -6 -2 z 2 a -8 + 3 z 2 - a 3 z + 5 a z + 8 z a -1 -2 z a -3 -3 z a -5 + z a -7 + 1- a z -1 - a -1 z -1$ (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 L11a340. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$