L11n228

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_17,9,18,22 X_9,21,10,20 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_19,5,20,4 X_5,19,6,18 X_21,17,22,16
Gauss code	{1, -2, 3, 9, -10, -6, 7, -8}, {-5, -1, 2, -3, 6, -7, 8, 11, -4, 10, -9, 5, -11, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 - v 3 u 2 + v 2 u 2 -2 v u 2 + u 2 + v 3 u -2 v 2 u + v u - u + v -1$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 7 -7 a z 5 + z 5 a -1 + a 5 z 3 + 2 a 3 z 3 -15 a z 3 + 5 z 3 a -1 + a 5 z + 4 a 3 z -13 a z + 6 z a -1 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 - z 8 a -2 -2 z 8 - a 5 z 7 + 8 a z 7 + 7 z 7 a -1 -2 a 6 z 6 -2 a 4 z 6 + 8 a 2 z 6 + 7 z 6 a -2 + 15 z 6 - a 7 z 5 + a 5 z 5 -2 a 3 z 5 -19 a z 5 -15 z 5 a -1 + 6 a 6 z 4 + 4 a 4 z 4 -18 a 2 z 4 -15 z 4 a -2 -31 z 4 + 3 a 7 z 3 + 3 a 5 z 3 + 6 a 3 z 3 + 20 a z 3 + 14 z 3 a -1 -3 a 6 z 2 + 12 a 2 z 2 + 11 z 2 a -2 + 20 z 2 -2 a 7 z -6 a 3 z -15 a z -7 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 =

-1 is the signature of L11n228. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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