L11n293

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u - v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$q 6 -3 q 5 + 4 q 4 -4 q 3 + 4 q 2 -2 q + 3 + q -1 - q -2 + 2 q -3 - q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 + z 6 - a 2 z 4 -5 z 4 a -2 + z 4 a -4 + 6 z 4 -3 a 2 z 2 -8 z 2 a -2 + 2 z 2 a -4 + 9 z 2 - a 2 -3 a -2 + a -4 + 3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$a z 9 + 2 z 9 a -1 + z 9 a -3 + 2 a 2 z 8 + 5 z 8 a -2 + 3 z 8 a -4 + 4 z 8 + a 3 z 7 -5 a z 7 -10 z 7 a -1 - z 7 a -3 + 3 z 7 a -5 -12 a 2 z 6 -30 z 6 a -2 -12 z 6 a -4 + z 6 a -6 -29 z 6 -5 a 3 z 5 + 3 z 5 a -1 -13 z 5 a -3 -11 z 5 a -5 + 19 a 2 z 4 + 50 z 4 a -2 + 10 z 4 a -4 -3 z 4 a -6 + 56 z 4 + 5 a 3 z 3 + 11 a z 3 + 19 z 3 a -1 + 20 z 3 a -3 + 7 z 3 a -5 -13 a 2 z 2 -30 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -36 z 2 - a 3 z -5 a z -9 z a -1 -7 z a -3 -2 z a -5 + 2 a 2 + 6 a -2 + 2 a -4 + 7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11n293. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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