L11n382

(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_5,10,6,11 X₈₄₉₃ X_22,18,19,17 X_20,12,21,11 X_12,20,13,19 X_18,22,5,21 X_9,16,10,17 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 $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 - v w u 2 + w u 2 -2 u 2 - v u + 2 v w u -2 w u + u - v w + w$ (db)
Jones polynomial	$2 q 4 -2 q 3 + 6 q 2 -5 q + 6-5 q -1 + 4 q -2 -2 q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2 z 4 -2 a 2 z 2 -4 z 2 a -2 + 6 z 2 -2 a 2 -8 a -2 + 2 a -4 + 8- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 3 z 7 a -1 + z 7 a -3 + a 2 z 6 -3 z 6 a -2 -2 z 6 -5 a z 5 -8 z 5 a -1 -3 z 5 a -3 + 2 a 2 z 4 + 11 z 4 a -2 + 3 z 4 a -4 + 10 z 4 + 3 a 3 z 3 + 13 a z 3 + 19 z 3 a -1 + 9 z 3 a -3 -4 a 2 z 2 -20 z 2 a -2 -9 z 2 a -4 -15 z 2 -3 a 3 z -14 a z -24 z a -1 -13 z a -3 + 2 a 2 + 16 a -2 + 8 a -4 + 11 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 L11n382. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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