L11n284

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_18,10,19,9 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_10,20,5,19
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {-8, -2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 - v 2 u 3 + v u 3 + v 2 w u 3 - v w u 3 + v u - v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$-2 q 7 + 3 q 6 -4 q 5 + 6 q 4 -4 q 3 + 6 q 2 -3 q + 3- q -1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 - z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -4 z 4 a -2 + 11 z 4 a -4 -5 z 4 a -6 -2 z 2 a -2 + 6 z 2 a -4 -5 z 2 a -6 + z 2 a -8 + 3 a -2 -5 a -4 + 2 a -6 + 2 a -2 z -2 -5 a -4 z -2 + 4 a -6 z -2 - a -8 z -2$ (db)
Kauffman polynomial	$2 z 9 a -3 + 2 z 9 a -5 + 3 z 8 a -2 + 7 z 8 a -4 + 4 z 8 a -6 + z 7 a -1 -7 z 7 a -3 -6 z 7 a -5 + 2 z 7 a -7 -15 z 6 a -2 -35 z 6 a -4 -20 z 6 a -6 -4 z 5 a -1 + z 5 a -3 -3 z 5 a -5 -8 z 5 a -7 + 21 z 4 a -2 + 49 z 4 a -4 + 29 z 4 a -6 + z 4 a -8 + 3 z 3 a -1 + 4 z 3 a -3 + 7 z 3 a -5 + 6 z 3 a -7 -9 z 2 a -2 -21 z 2 a -4 -14 z 2 a -6 -2 z 2 a -8 + 5 z a -3 + 9 z a -5 + 5 z a -7 + z a -9 -3 a -2 -4 a -4 -2 a -6 -5 a -3 z -1 -9 a -5 z -1 -5 a -7 z -1 - a -9 z -1 + 2 a -2 z -2 + 5 a -4 z -2 + 4 a -6 z -2 + a -8 z -2$ (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 =

4 is the signature of L11n284. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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