L11n209

(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_5,14,6,15 X_22,18,9,17 X_19,5,20,4 X_6,22,7,21 X_16,7,17,8 X_8,9,1,10 X_13,18,14,19 X_20,15,21,16
Gauss code	{1, -2, 3, 6, -4, -7, 8, -9}, {9, -1, 2, -3, -10, 4, 11, -8, 5, 10, -6, -11, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 + v 3 u 2 -4 v 2 u 2 + 2 v u 2 + 2 v 2 u -4 v u + u + v -1$ (db)
Jones polynomial	$\frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 4 z 3 a 7 + 4 z a 7 - z 7 a 5 -6 z 5 a 5 -12 z 3 a 5 -7 z a 5 + a 5 z -1 + z 5 a 3 + 3 z 3 a 3 + z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 - z a 11 -2 z 6 a 10 + 6 z 4 a 10 -3 z 2 a 10 -2 z 7 a 9 + 5 z 5 a 9 -2 z 3 a 9 -2 z 8 a 8 + 7 z 6 a 8 -10 z 4 a 8 + 5 z 2 a 8 - z 9 a 7 + 3 z 7 a 7 -5 z 5 a 7 + 3 z 3 a 7 -3 z a 7 -3 z 8 a 6 + 13 z 6 a 6 -24 z 4 a 6 + 12 z 2 a 6 - z 9 a 5 + 5 z 7 a 5 -14 z 5 a 5 + 15 z 3 a 5 -6 z a 5 - a 5 z -1 - z 8 a 4 + 4 z 6 a 4 -9 z 4 a 4 + 6 z 2 a 4 + a 4 -3 z 5 a 3 + 7 z 3 a 3 -2 z a 3 - a 3 z -1 - z 4 a 2 + 2 z 2 a 2$ (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 =

-3 is the signature of L11n209. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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