L11n384

(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_10,6,11,5 X₈₄₉₃ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {4, -1, 2, -5, 10, -4, -7, 8, 11, -2, 3, -10, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -3 v u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 6 -3 q 5 + 5 q 4 -8 q 3 + 11 q 2 -10 q + 11-7 q -1 + 6 q -2 -2 q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 -4 z 4 a -2 + z 4 a -4 + 3 z 4 -2 a 2 z 2 -7 z 2 a -2 + 2 z 2 a -4 + 7 z 2 - a 2 -3 a -2 + a -4 + 3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 z 9 a -1 + 2 z 9 a -3 + 9 z 8 a -2 + 4 z 8 a -4 + 5 z 8 + 4 a z 7 + z 7 a -1 + 3 z 7 a -5 + a 2 z 6 -31 z 6 a -2 -13 z 6 a -4 + z 6 a -6 -16 z 6 -7 a z 5 -13 z 5 a -1 -16 z 5 a -3 -10 z 5 a -5 + 6 a 2 z 4 + 41 z 4 a -2 + 12 z 4 a -4 -3 z 4 a -6 + 32 z 4 + 3 a 3 z 3 + 10 a z 3 + 20 z 3 a -1 + 21 z 3 a -3 + 8 z 3 a -5 -9 a 2 z 2 -26 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -28 z 2 - a 3 z -5 a z -9 z a -1 -7 z a -3 -2 z a -5 + 2 a 2 + 6 a -2 + 2 a -4 + 7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11n384. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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