L11n392

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₈₄₉₃ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_11,18,12,5 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 - v w u 3 + w u 3 -2 u 3 -5 v u 2 + 4 v w u 2 -4 w u 2 + 5 u 2 + 4 v u -5 v w u + 5 w u -4 u - v + 2 v w -2 w + 1$ (db)
Jones polynomial	$3 q 4 -6 q 3 + 13 q 2 -14 q + 17-16 q -1 + 13 q -2 -9 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + 2 a 2 z 4 + z 4 a -2 - z 4 - a 4 z 2 + a 2 z 2 -3 z 2 a -2 + 3 z 2 -2 a 2 -8 a -2 + 2 a -4 + 8- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 7 a 2 z 8 + 5 z 8 a -2 + 12 z 8 + 8 a 3 z 7 + 15 a z 7 + 10 z 7 a -1 + 3 z 7 a -3 + 4 a 4 z 6 -6 a 2 z 6 -8 z 6 a -2 -18 z 6 + a 5 z 5 -14 a 3 z 5 -38 a z 5 -26 z 5 a -1 -3 z 5 a -3 -5 a 4 z 4 -4 a 2 z 4 + 17 z 4 a -2 + 6 z 4 a -4 + 12 z 4 - a 5 z 3 + 9 a 3 z 3 + 29 a z 3 + 29 z 3 a -1 + 10 z 3 a -3 + 2 a 4 z 2 -24 z 2 a -2 -11 z 2 a -4 -15 z 2 -3 a 3 z -14 a z -24 z a -1 -13 z a -3 + 2 a 2 + 16 a -2 + 8 a -4 + 11 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 =

0 is the signature of L11n392. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$