L11n431

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,10,21,9 X_5,15,6,14 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_10,4,11,3 X_18,11,19,12 X_15,1,16,6 X_4,20,5,19 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 $u$ , $v$ , $w$ , ...)	$- u v 3 + u w v 3 - w v 3 + v 3 - u 2 w 2 v 2 + u w 2 v 2 + 2 u v 2 + u 2 w v 2 -4 u w v 2 + 2 w v 2 -2 v 2 + 2 u 2 w 2 v -2 u w 2 v - u v -2 u 2 w v + 4 u w v - w v + v - u 2 w 2 + u w 2 + u 2 w - u w$ (db)
Jones polynomial	$q 9 -4 q 8 + 7 q 7 -9 q 6 + 12 q 5 -11 q 4 + 11 q 3 -7 q 2 + 5 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -4 - z 4 a -2 + 3 z 4 a -4 -2 z 4 a -6 + 3 z 2 a -4 -3 z 2 a -6 + z 2 a -8 + 2 a -2 -2 a -4 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$3 z 9 a -5 + 3 z 9 a -7 + 5 z 8 a -4 + 11 z 8 a -6 + 6 z 8 a -8 + 2 z 7 a -3 -5 z 7 a -5 -3 z 7 a -7 + 4 z 7 a -9 -16 z 6 a -4 -36 z 6 a -6 -19 z 6 a -8 + z 6 a -10 + z 5 a -5 -10 z 5 a -7 -11 z 5 a -9 + 5 z 4 a -2 + 25 z 4 a -4 + 37 z 4 a -6 + 15 z 4 a -8 -2 z 4 a -10 + z 3 a -1 + 2 z 3 a -5 + 8 z 3 a -7 + 5 z 3 a -9 -3 z 2 a -2 -10 z 2 a -4 -11 z 2 a -6 -4 z 2 a -8 + 2 z a -3 + 2 z a -5 -2 a -2 -3 a -4 -2 a -6 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L11n431. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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