L11n229

(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_20,5,21,6 X_7,14,8,15 X_13,16,14,17 X_15,8,16,1 X_22,17,9,18 X_18,21,19,22 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, -5, 7}, {10, -1, 2, -3, -6, 5, -7, 6, 8, -9, 11, -4, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + v 2 u 3 + v 3 u 2 -2 v 2 u 2 -2 v u + u + v -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{2}{q^{23/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 + z 5 a 11 + 5 z 3 a 11 + 5 z a 11 - z 7 a 9 -5 z 5 a 9 -6 z 3 a 9 - z a 9 + a 9 z -1 - z 7 a 7 -6 z 5 a 7 -11 z 3 a 7 -7 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$-2 z a 15 - z 4 a 14 - z 7 a 13 + 3 z 5 a 13 -2 z 3 a 13 - z a 13 -2 z 8 a 12 + 10 z 6 a 12 -17 z 4 a 12 + 11 z 2 a 12 - z 9 a 11 + 4 z 7 a 11 -6 z 5 a 11 + 9 z 3 a 11 -5 z a 11 -3 z 8 a 10 + 14 z 6 a 10 -18 z 4 a 10 + 8 z 2 a 10 - z 9 a 9 + 4 z 7 a 9 -3 z 5 a 9 + z a 9 - a 9 z -1 - z 8 a 8 + 4 z 6 a 8 -2 z 4 a 8 -3 z 2 a 8 + a 8 - z 7 a 7 + 6 z 5 a 7 -11 z 3 a 7 + 7 z a 7 - a 7 z -1$ (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 =

-7 is the signature of L11n229. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$
$r = -8$	${\mathbb Z}^{2}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$