L11n393

(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_12,6,13,5 X₈₄₉₃ X_13,22,14,19 X_9,20,10,21 X_19,10,20,11 X_21,14,22,15 X_18,12,5,11 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 $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -2 u 4 - v u 3 + u 3 - v w u 2 + w u 2 + 2 v w u -2 w u - v w + w$ (db)
Jones polynomial	$q 3 - q 2 - q + 3-4 q -1 + 6 q -2 -5 q -3 + 7 q -4 -4 q -5 + 3 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 + a 6 z -2 - a 6 + 2 z 4 a 4 + 5 z 2 a 4 -2 a 4 z -2 + a 4 - z 6 a 2 -4 z 4 a 2 -4 z 2 a 2 + a 2 z -2 - z 2 -1 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$2 a 5 z 9 + 2 a 3 z 9 + 3 a 6 z 8 + 8 a 4 z 8 + 5 a 2 z 8 + a 7 z 7 -5 a 5 z 7 -4 a 3 z 7 + 3 a z 7 + z 7 a -1 -14 a 6 z 6 -39 a 4 z 6 -27 a 2 z 6 + z 6 a -2 - z 6 -4 a 7 z 5 -7 a 5 z 5 -15 a 3 z 5 -19 a z 5 -7 z 5 a -1 + 19 a 6 z 4 + 55 a 4 z 4 + 41 a 2 z 4 -5 z 4 a -2 + 4 a 7 z 3 + 14 a 5 z 3 + 30 a 3 z 3 + 30 a z 3 + 10 z 3 a -1 -10 a 6 z 2 -31 a 4 z 2 -23 a 2 z 2 + 4 z 2 a -2 + 2 z 2 - a 7 z -3 a 5 z -11 a 3 z -13 a z -4 z a -1 + 4 a 4 + 5 a 2 - a -2 + 1-2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

0 is the signature of L11n393. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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