L11n419

(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_2,18,3,17 X_9,19,10,18 X_20,12,21,11 X_5,14,6,15 X_15,13,16,22 X_13,6,14,1 X_19,5,20,4 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 $u$ , $v$ , $w$ , ...)	$- u v 3 + u w v 3 + u v 2 + u 2 w v 2 -2 u w v 2 - u w 2 v + 2 u w v - w v + u w 2 - u w$ (db)
Jones polynomial	$- q 6 + 3 q 5 -3 q 4 + 5 q 3 -4 q 2 + 4 q -2 + 2 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -2 -7 z 2 a -2 + 3 z 2 a -4 + 2 z 2 -8 a -2 + 5 a -4 - a -6 + 4-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 a -4 + z 7 a -1 + 2 z 7 a -3 + z 7 a -5 -5 z 6 a -2 -5 z 6 a -4 -3 z 5 a -1 -6 z 5 a -3 -3 z 5 a -5 + 16 z 4 a -2 + 16 z 4 a -4 + 3 z 4 a -6 + 3 z 4 + 5 z 3 a -1 + 11 z 3 a -3 + 7 z 3 a -5 + z 3 a -7 -23 z 2 a -2 -18 z 2 a -4 -4 z 2 a -6 -9 z 2 -6 z a -1 -8 z a -3 -3 z a -5 - z a -7 + 12 a -2 + 8 a -4 + a -6 + 6 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11n419. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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