L11a431

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_18,10,19,9 X_20,11,21,12 X_22,20,13,19 X_12,14,5,13 X_4,17,1,18 X_10,21,11,22
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11, 7, -9}, {9, -2, 4, -5, 10, -6, 8, -7, 11, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 + v u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 4 v 2 u 2 -3 v u 2 + 2 v 3 w u 2 -4 v 2 w u 2 + 3 v w u 2 - w u 2 + u 2 + v 3 u -3 v 2 u + 4 v u - v 3 w u + 3 v 2 w u -4 v w u + 2 w u -2 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$q 7 -4 q 6 + 7 q 5 -11 q 4 + 16 q 3 -17 q 2 + 18 q -14 + 12 q -1 -7 q -2 + 4 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -8 z 4 a -2 + 3 z 4 a -4 + 7 z 4 -2 a 2 z 2 -3 z 2 a -2 + z 2 a -4 + 4 z 2 + a 2 + 3 a -2 - a -4 -3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 5 a z 9 + 11 z 9 a -1 + 6 z 9 a -3 + 4 a 2 z 8 + 9 z 8 a -2 + 8 z 8 a -4 + 5 z 8 + a 3 z 7 -16 a z 7 -31 z 7 a -1 -6 z 7 a -3 + 8 z 7 a -5 -15 a 2 z 6 -36 z 6 a -2 -8 z 6 a -4 + 7 z 6 a -6 -36 z 6 -3 a 3 z 5 + 11 a z 5 + 19 z 5 a -1 -5 z 5 a -3 -6 z 5 a -5 + 4 z 5 a -7 + 16 a 2 z 4 + 28 z 4 a -2 -4 z 4 a -4 -7 z 4 a -6 + z 4 a -8 + 40 z 4 + 2 a 3 z 3 + 2 z 3 a -3 -3 z 3 a -5 -3 z 3 a -7 -5 a 2 z 2 + z 2 a -2 + 5 z 2 a -4 + 2 z 2 a -6 -7 z 2 + a z + 3 z a -1 + 3 z a -3 + z a -5 -2 a 2 -6 a -2 -2 a -4 -5-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11a431. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$