L11n327

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_13,18,14,19 X_9,21,10,20 X_19,5,20,12 X_11,13,12,22 X_21,11,22,10 X_17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, 4, -5, -11, 6, -8, 7, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 + v 3 u 2 -2 v 2 u 2 + 2 v 2 w u 2 -3 v w u 2 + w u 2 - v 3 u + 3 v 2 u -2 v u + 2 v w u - w u - v 2 + v$ (db)
Jones polynomial	$2 q 5 -4 q 4 + 6 q 3 -6 q 2 + 8 q -6 + 6 q -1 -3 q -2 + 2 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 -3 z 4 a -2 + 5 z 4 -3 a 2 z 2 -10 z 2 a -2 + 2 z 2 a -4 + 10 z 2 -2 a 2 -9 a -2 + 3 a -4 + 8-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + 2 a 2 z 8 + 3 z 8 a -2 + 5 z 8 + a 3 z 7 - a z 7 + 2 z 7 a -1 + 4 z 7 a -3 -10 a 2 z 6 -8 z 6 a -2 + 3 z 6 a -4 -21 z 6 -5 a 3 z 5 -9 a z 5 -15 z 5 a -1 -10 z 5 a -3 + z 5 a -5 + 16 a 2 z 4 + 12 z 4 a -2 -3 z 4 a -4 + 31 z 4 + 7 a 3 z 3 + 16 a z 3 + 18 z 3 a -1 + 12 z 3 a -3 + 3 z 3 a -5 -10 a 2 z 2 -19 z 2 a -2 + 3 z 2 a -6 -26 z 2 -2 a 3 z -7 a z -12 z a -1 -8 z a -3 - z a -5 + 3 a 2 + 11 a -2 + 3 a -4 - a -6 + 11 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11n327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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