L11a428

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 -2 v u 4 - v 2 w u 4 + 2 v w u 4 + u 4 -2 v 2 u 3 + 4 v u 3 + 2 v 2 w u 3 -5 v w u 3 + 2 w u 3 -2 u 3 + 2 v 2 u 2 -5 v u 2 -2 v 2 w u 2 + 5 v w u 2 -2 w u 2 + 2 u 2 -2 v 2 u + 5 v u + 2 v 2 w u -4 v w u + 2 w u -2 u -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 14 q 2 -19 q + 22-20 q -1 + 19 q -2 -12 q -3 + 8 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -8 a 2 z 4 -3 z 4 a -2 + 10 z 4 + 3 a 4 z 2 -12 a 2 z 2 -3 z 2 a -2 + 11 z 2 + 3 a 4 -10 a 2 -2 a -2 + 9 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 5 a 3 z 9 + 12 a z 9 + 7 z 9 a -1 + 6 a 4 z 8 + 10 a 2 z 8 + 10 z 8 a -2 + 14 z 8 + 3 a 5 z 7 -6 a 3 z 7 -23 a z 7 -6 z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -18 a 4 z 6 -41 a 2 z 6 -19 z 6 a -2 + 4 z 6 a -4 -45 z 6 -7 a 5 z 5 -9 a 3 z 5 + 5 a z 5 -7 z 5 a -1 -13 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 24 a 4 z 4 + 59 a 2 z 4 + 16 z 4 a -2 -5 z 4 a -4 + 53 z 4 + 3 a 5 z 3 + 20 a 3 z 3 + 26 a z 3 + 16 z 3 a -1 + 6 z 3 a -3 - z 3 a -5 + 2 a 6 z 2 -22 a 4 z 2 -46 a 2 z 2 -8 z 2 a -2 -30 z 2 -16 a 3 z -27 a z -13 z a -1 -2 z a -3 + 10 a 4 + 20 a 2 + 2 a -2 + 13 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

0 is the signature of L11a428. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$