L11a438

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_10,4,11,3 X_14,6,15,5 X_22,12,13,11 X_12,14,5,13 X_4,21,1,22 X_20,17,21,18 X_16,7,17,8 X_8,20,9,19 X_18,10,19,9
Gauss code	{1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, {6, -4, 2, -9, 8, -11, 10, -8, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 + v u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 5 v 2 u 2 -4 v u 2 + 2 v 3 w u 2 -5 v 2 w u 2 + 4 v w u 2 - w u 2 + u 2 + v 3 u -4 v 2 u + 5 v u - v 3 w u + 4 v 2 w u -5 v w u + 2 w u -2 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$- q 8 + 4 q 7 -8 q 6 + 14 q 5 -17 q 4 + 21 q 3 -20 q 2 + 18 q -12 + 8 q -1 -4 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -5 z 6 a -2 + 2 z 6 a -4 + z 6 -9 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 + 3 z 4 -6 z 2 a -2 + 6 z 2 a -4 -2 z 2 a -6 + 2 z 2 - a -4 + 1 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 a -4 + 6 z 9 a -1 + 12 z 9 a -3 + 6 z 9 a -5 + 10 z 8 a -2 + 11 z 8 a -4 + 8 z 8 a -6 + 7 z 8 + 4 a z 7 -12 z 7 a -1 -28 z 7 a -3 -5 z 7 a -5 + 7 z 7 a -7 + a 2 z 6 -41 z 6 a -2 -35 z 6 a -4 -11 z 6 a -6 + 4 z 6 a -8 -20 z 6 -10 a z 5 + 4 z 5 a -1 + 24 z 5 a -3 - z 5 a -5 -10 z 5 a -7 + z 5 a -9 -2 a 2 z 4 + 47 z 4 a -2 + 37 z 4 a -4 + 3 z 4 a -6 -6 z 4 a -8 + 17 z 4 + 4 a z 3 + z 3 a -1 -7 z 3 a -3 - z 3 a -5 + 2 z 3 a -7 - z 3 a -9 -16 z 2 a -2 -11 z 2 a -4 + z 2 a -6 + 2 z 2 a -8 -6 z 2 + 4 z a -3 + 4 z a -5 - a -2 -4 a -4 -3 a -6 + 1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L11a438. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$