L11a289

(Knotscape image)

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

Celtic or pseudo-Celtic linear decorative knot

Decorative variant with big loops at ends

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_20,14,21,13 X_14,7,15,8 X_18,16,19,15 X_16,6,17,5 X_6,18,7,17 X_4,19,5,20 X_8,22,1,21
Gauss code	{1, -4, 2, -10, 8, -9, 6, -11}, {4, -1, 3, -2, 5, -6, 7, -8, 9, -7, 10, -5, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -3 v u 3 + u 3 + 3 v 3 u 2 -7 v 2 u 2 + 7 v u 2 -3 u 2 -3 v 3 u + 7 v 2 u -7 v u + 3 u + v 3 -3 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-15 q^{3/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -6 z 3 a -1 + 7 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + 2 a z -5 z a -1 + 5 z a -3 -3 z a -5 + z a -7 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -8 z 8 a -2 -13 z 8 a -4 -8 z 8 a -6 -3 z 8 - a z 7 + 5 z 7 a -1 + 7 z 7 a -3 -9 z 7 a -5 -10 z 7 a -7 + 32 z 6 a -2 + 33 z 6 a -4 + 4 z 6 a -6 -8 z 6 a -8 + 11 z 6 + 4 a z 5 + 10 z 5 a -1 + 24 z 5 a -3 + 34 z 5 a -5 + 12 z 5 a -7 -4 z 5 a -9 -28 z 4 a -2 -16 z 4 a -4 + 9 z 4 a -6 + 8 z 4 a -8 - z 4 a -10 -12 z 4 -6 a z 3 -22 z 3 a -1 -32 z 3 a -3 -24 z 3 a -5 -6 z 3 a -7 + 2 z 3 a -9 + 4 z 2 a -2 -4 z 2 a -6 -3 z 2 a -8 + 3 z 2 + 4 a z + 10 z a -1 + 10 z a -3 + 6 z a -5 + 2 z a -7 + 1- a z -1 - a -1 z -1$ (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 =

3 is the signature of L11a289. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$