L11n256

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_16,12,17,11 X_18,22,19,21 X_20,14,21,13 X_12,20,13,19 X_22,18,9,17 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, 3, -6, 5, 9, -8, -3, 7, -4, 6, -5, 4, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 -2 v u 2 - v w u 2 - w u 2 - u 2 + v u + v w u + 2 w u + u -2 w$ (db)
Jones polynomial	$- q 7 + q 6 - q 5 - q 4 + 2 q 3 - q 2 + 4 q -2 + 3 q -1 - q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 4 a -2 + z 4 a -4 - z 4 + a 2 z 2 -3 z 2 a -2 + 5 z 2 a -4 - z 2 a -6 -3 z 2 + 2 a 2 -3 a -2 + 6 a -4 -2 a -6 -3 + a 2 z -2 -2 a -2 z -2 + 3 a -4 z -2 - a -6 z -2 - z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 a -4 + z 8 a -6 + z 8 + a z 7 + 3 z 7 a -1 + 3 z 7 a -3 + 2 z 7 a -5 + z 7 a -7 + a 2 z 6 -5 z 6 a -2 -7 z 6 a -4 -6 z 6 a -6 -3 z 6 -3 a z 5 -14 z 5 a -1 -21 z 5 a -3 -16 z 5 a -5 -6 z 5 a -7 -5 a 2 z 4 + 8 z 4 a -2 + 12 z 4 a -4 + 8 z 4 a -6 - z 4 - a z 3 + 18 z 3 a -1 + 44 z 3 a -3 + 35 z 3 a -5 + 10 z 3 a -7 + 7 a 2 z 2 -8 z 2 a -2 -7 z 2 a -4 -2 z 2 a -6 + 4 z 2 + 4 a z -10 z a -1 -34 z a -3 -27 z a -5 -7 z a -7 -4 a 2 + 4 a -2 + 5 a -4 + a -6 -3-2 a z -1 + 2 a -1 z -1 + 10 a -3 z -1 + 8 a -5 z -1 + 2 a -7 z -1 + a 2 z -2 -2 a -2 z -2 -3 a -4 z -2 - a -6 z -2 + z -2$ (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 =

2 is the signature of L11n256. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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