L11n321

(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_9,18,10,19 X_17,1,18,4 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X_11,21,12,20
Gauss code	{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 + v 3 u 2 -2 v 2 u 2 + v u 2 + 2 v 2 w u 2 -4 v w u 2 + w u 2 - v 3 u + 4 v 2 u -2 v u - v 2 w u + 2 v w u - w u - v 2 + v$ (db)
Jones polynomial	$-2 q 3 + 5 q 2 -6 q + 9-8 q -1 + 9 q -2 -6 q -3 + 4 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- a 2 z 6 + a 4 z 4 -5 a 2 z 4 + 3 z 4 + 3 a 4 z 2 -11 a 2 z 2 -2 z 2 a -2 + 9 z 2 + 3 a 4 -9 a 2 -2 a -2 + 8 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 2 a 4 z 8 + 5 a 2 z 8 + 3 z 8 + 2 a 5 z 7 + a 3 z 7 + 2 a z 7 + 3 z 7 a -1 + a 6 z 6 -5 a 4 z 6 -16 a 2 z 6 + z 6 a -2 -9 z 6 -7 a 5 z 5 -10 a 3 z 5 -9 a z 5 -6 z 5 a -1 -4 a 6 z 4 + 2 a 4 z 4 + 24 a 2 z 4 + 4 z 4 a -2 + 22 z 4 + 6 a 5 z 3 + 13 a 3 z 3 + 15 a z 3 + 11 z 3 a -1 + 3 z 3 a -3 + 4 a 6 z 2 -2 a 4 z 2 -23 a 2 z 2 -7 z 2 a -2 -24 z 2 - a 5 z -8 a 3 z -12 a z -7 z a -1 -2 z a -3 - a 6 + 3 a 4 + 11 a 2 + 3 a -2 + 11 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11n321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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