L11a429

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,12,19,11 X_16,13,17,14 X_14,6,15,5 X_10,16,5,15 X_22,20,11,19 X_8,22,9,21 X_20,8,21,7 X_2,9,3,10 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {5, -1, 9, -8, 10, -6}, {3, -2, 4, -5, 6, -4, 11, -3, 7, -9, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -4 v u 2 -2 v 2 w u 2 + 5 v w u 2 -2 w u 2 + 2 u 2 -5 v 2 u + 9 v u + 4 v 2 w u -9 v w u + 5 w u -4 u + 2 v 2 -5 v -2 v 2 w + 4 v w -2 w + 2$ (db)
Jones polynomial	$q 9 -4 q 8 + 9 q 7 -14 q 6 + 20 q 5 -22 q 4 + 23 q 3 -19 q 2 + 15 q -8 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 -2 z 4 a -6 - z 4 + 4 z 2 a -2 -4 z 2 a -4 - z 2 a -6 + z 2 a -8 - z 2 + 4 a -2 -6 a -4 + 2 a -6 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 6 z 9 a -3 + 12 z 9 a -5 + 6 z 9 a -7 + 8 z 8 a -2 + 13 z 8 a -4 + 12 z 8 a -6 + 7 z 8 a -8 + 7 z 7 a -1 -18 z 7 a -5 -7 z 7 a -7 + 4 z 7 a -9 -7 z 6 a -2 -28 z 6 a -4 -34 z 6 a -6 -16 z 6 a -8 + z 6 a -10 + 4 z 6 + a z 5 -9 z 5 a -1 -9 z 5 a -3 + 6 z 5 a -5 -4 z 5 a -7 -9 z 5 a -9 -4 z 4 a -2 + 12 z 4 a -4 + 23 z 4 a -6 + 11 z 4 a -8 -2 z 4 a -10 -6 z 4 - a z 3 + 3 z 3 a -1 + z 3 a -3 -5 z 3 a -5 + 3 z 3 a -7 + 5 z 3 a -9 + 8 z 2 a -2 + 8 z 2 a -4 -2 z 2 a -6 -4 z 2 a -8 + z 2 a -10 + 3 z 2 + 6 z a -3 + 6 z a -5 -5 a -2 -8 a -4 -3 a -6 + a -8 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

2 is the signature of L11a429. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$