L11n428

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_5,15,6,14 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_15,1,16,6 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 2 v 2 - u 2 w v 2 -2 u v + 2 u w v - w + 1$ (db)
Jones polynomial	$- q 4 + q 3 + q + 1 + q -2 + q -4$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 - z 4 a -2 + 6 z 4 -6 a 2 z 2 -4 z 2 a -2 + 10 z 2 + 2 a 4 -7 a 2 -2 a -2 + 7 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + z 7 a -1 + z 7 a -3 -2 a 2 z 6 -7 z 6 a -2 -9 z 6 - a z 5 -7 z 5 a -1 -6 z 5 a -3 + a 4 z 4 + 13 a 2 z 4 + 15 z 4 a -2 + 27 z 4 + a 3 z 3 + 5 a z 3 + 13 z 3 a -1 + 9 z 3 a -3 -4 a 4 z 2 -24 a 2 z 2 -12 z 2 a -2 -32 z 2 -3 a 3 z -7 a z -6 z a -1 -2 z a -3 + 5 a 4 + 13 a 2 + 4 a -2 + 13 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

-1 is the signature of L11n428. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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