L11n421

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_17,2,18,3 X_9,19,10,18 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_4,19,5,20 X_12,22,7,21
Gauss code	{1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v 2 w u 3 + v 2 u 2 -2 v 2 w u 2 + v w u 2 - w 2 u - v w u + 2 w u + w 2 - w$ (db)
Jones polynomial	$q 5 - q 4 + 3 q 3 -3 q 2 + 4 q -3 + 4 q -1 -2 q -2 + 2 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 -2 z 4 a -2 + 5 z 4 -3 a 2 z 2 -8 z 2 a -2 + z 2 a -4 + 8 z 2 - a 2 -8 a -2 + 3 a -4 + 6-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + 2 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + a 3 z 7 -3 a z 7 -3 z 7 a -1 + z 7 a -3 -11 a 2 z 6 -11 z 6 a -2 -22 z 6 -5 a 3 z 5 -3 a z 5 -2 z 5 a -1 -4 z 5 a -3 + 18 a 2 z 4 + 23 z 4 a -2 + 2 z 4 a -4 + 39 z 4 + 6 a 3 z 3 + 9 a z 3 + 9 z 3 a -1 + 7 z 3 a -3 + z 3 a -5 -11 a 2 z 2 -25 z 2 a -2 -4 z 2 a -4 + z 2 a -6 -31 z 2 - a 3 z -3 a z -8 z a -1 -6 z a -3 + 2 a 2 + 12 a -2 + 4 a -4 - a -6 + 10 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11n421. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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