L11n333

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₃₈₄₉ X_2,18,3,17 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_19,1,20,4
Gauss code	{1, -4, -3, 11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, -11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 3 + w u 3 + v 2 u 2 - v u 2 - v 2 w u 2 + 3 v w u 2 - w u 2 + v 3 u -3 v 2 u + v u + v 2 w u - v w u - v 3 + v 2$ (db)
Jones polynomial	$q 2 -2 q + 5-5 q -1 + 6 q -2 -5 q -3 + 6 q -4 -3 q -5 + 2 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + z 4 a 4 + 2 z 2 a 4 + a 4 z -2 + 3 a 4 + z 4 a 2 -2 a 2 z -2 -3 a 2 -2 z 2 + z -2 + a -2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 5 a 4 z 8 + 3 a 2 z 8 + a 7 z 7 - a 5 z 7 + a 3 z 7 + 3 a z 7 -10 a 6 z 6 -23 a 4 z 6 -12 a 2 z 6 + z 6 -5 a 7 z 5 -10 a 5 z 5 -16 a 3 z 5 -11 a z 5 + 16 a 6 z 4 + 36 a 4 z 4 + 19 a 2 z 4 - z 4 + 7 a 7 z 3 + 17 a 5 z 3 + 23 a 3 z 3 + 15 a z 3 + 2 z 3 a -1 -11 a 6 z 2 -31 a 4 z 2 -20 a 2 z 2 + z 2 a -2 + z 2 -2 a 7 z -7 a 5 z -12 a 3 z -8 a z - z a -1 + 3 a 6 + 11 a 4 + 11 a 2 - a -2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n333. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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