L11n331

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_13,18,14,19 X_9,13,10,22 X_11,21,12,20 X_19,5,20,12 X_21,11,22,10 X_4,17,1,18
Gauss code	{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -8, 9}, {-6, 2, 4, -5, 11, 6, -9, 8, -10, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 2 - v w u 2 + w u 2 - u 2 + v 2 u -2 v u - v 2 w u + 2 v w u - w u + u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$q 4 - q 3 + 4 q 2 -4 q + 6-5 q -1 + 5 q -2 -3 q -3 + 2 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 4 - a 4 + z 4 a 2 + 2 z 2 a 2 + 2 a 2 + z 4 + z 2 + z -2 + 1-2 z 2 a -2 -2 a -2 z -2 -3 a -2 + a -4 z -2 + a -4$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + 2 a 2 z 8 + z 8 a -2 + 3 z 8 + 2 a 3 z 7 -3 a z 7 -5 z 7 a -1 + 2 a 4 z 6 -7 a 2 z 6 -6 z 6 a -2 -15 z 6 + a 5 z 5 -5 a 3 z 5 + 2 a z 5 + 9 z 5 a -1 + z 5 a -3 -6 a 4 z 4 + 8 a 2 z 4 + 17 z 4 a -2 + z 4 a -4 + 30 z 4 -3 a 5 z 3 + a 3 z 3 + 5 a z 3 - z 3 a -3 + 3 a 4 z 2 -2 a 2 z 2 -20 z 2 a -2 -4 z 2 a -4 -21 z 2 + a 5 z + a 3 z -3 a z -6 z a -1 -3 z a -3 - a 4 + 9 a -2 + 4 a -4 + 7 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

0 is the signature of L11n331. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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