L11n292

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_11,18,12,19 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,11,10,22 X_17,1,18,4 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 $u$ , $v$ , $w$ , ...)	$- v w u 4 + w u 4 + v 2 u 3 - v u 3 + 2 v w u 3 -2 w u 3 -2 v 2 u 2 + v u 2 - v w u 2 + 2 w u 2 + 2 v 2 u -2 v u + v w u - w u - v 2 + v$ (db)
Jones polynomial	$q 5 -3 q 4 + 5 q 3 -6 q 2 + 8 q -6 + 7 q -1 -4 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 -2 z 4 a -2 + 4 z 4 -2 a 2 z 2 -5 z 2 a -2 + z 2 a -4 + 5 z 2 -2 a -2 + a -4 + 1 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 3 a 2 z 8 + 5 z 8 a -2 + 8 z 8 + a 3 z 7 -5 a z 7 -2 z 7 a -1 + 4 z 7 a -3 -14 a 2 z 6 -21 z 6 a -2 + z 6 a -4 -36 z 6 -4 a 3 z 5 -6 a z 5 -16 z 5 a -1 -14 z 5 a -3 + 19 a 2 z 4 + 30 z 4 a -2 + z 4 a -4 + 48 z 4 + 4 a 3 z 3 + 14 a z 3 + 24 z 3 a -1 + 17 z 3 a -3 + 3 z 3 a -5 -9 a 2 z 2 -20 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -25 z 2 - a 3 z -3 a z -7 z a -1 -7 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11n292. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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