L11n396

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_9,22,10,19 X₈₄₉₃ X_21,17,22,16 X_11,5,12,18 X_5,21,6,20 X_17,11,18,10 X_19,12,20,13 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, -5, -4, 9, -7, 10, 11, -2, 3, 6, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q 6 -2 q 5 + 2 q 4 -2 q 3 + q 2 + q + 1 + 3 q -1 -2 q -2 + 2 q -3 - q -4$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 6 a -2 + z 6 - a 2 z 4 -6 z 4 a -2 + z 4 a -4 + 6 z 4 -3 a 2 z 2 -11 z 2 a -2 + 3 z 2 a -4 + 11 z 2 -2 a 2 -6 a -2 + 2 a -4 + 6 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$a z 9 + 2 z 9 a -1 + z 9 a -3 + 2 a 2 z 8 + 5 z 8 a -2 + 2 z 8 a -4 + 5 z 8 + a 3 z 7 -3 a z 7 -9 z 7 a -1 -3 z 7 a -3 + 2 z 7 a -5 -11 a 2 z 6 -33 z 6 a -2 -10 z 6 a -4 + z 6 a -6 -33 z 6 -5 a 3 z 5 -8 a z 5 -4 z 5 a -1 -10 z 5 a -3 -9 z 5 a -5 + 17 a 2 z 4 + 59 z 4 a -2 + 11 z 4 a -4 -4 z 4 a -6 + 61 z 4 + 6 a 3 z 3 + 20 a z 3 + 32 z 3 a -1 + 26 z 3 a -3 + 8 z 3 a -5 -14 a 2 z 2 -40 z 2 a -2 -8 z 2 a -4 + 2 z 2 a -6 -44 z 2 -2 a 3 z -10 a z -18 z a -1 -14 z a -3 -4 z a -5 + 4 a 2 + 12 a -2 + 4 a -4 + 13-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

1 is the signature of L11n396. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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