L11n221

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_12,4,13,3 X_22,16,9,15 X_2,17,3,18 X_21,4,22,5 X_5,15,6,14 X_13,21,14,20 X_16,12,17,11 X_6,19,7,20 X_18,7,19,8
Gauss code	{1, -5, 3, 6, -7, -10, 11, -2}, {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 -2 v 3 u 2 + 6 v 2 u 2 -6 v u 2 + u 2 + v 3 u -6 v 2 u + 6 v u -2 u + v 2 - v$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2 a z 5 - z 5 a -1 + 2 a 3 z 3 -6 a z 3 - z 3 a -1 + z 3 a -3 + 3 a 3 z -7 a z + z a -1 + z a -3 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -4 a 2 z 8 -4 z 8 a -2 -8 z 8 -3 a 3 z 7 -3 z 7 a -3 - a 4 z 6 + 7 a 2 z 6 + 11 z 6 a -2 - z 6 a -4 + 20 z 6 - a z 5 + 8 z 5 a -1 + 9 z 5 a -3 -4 a 4 z 4 -11 a 2 z 4 -8 z 4 a -2 + 3 z 4 a -4 -18 z 4 -3 a 5 z 3 + 3 a 3 z 3 + 11 a z 3 -2 z 3 a -1 -7 z 3 a -3 + 2 a 4 z 2 + 9 a 2 z 2 + 3 z 2 a -2 -2 z 2 a -4 + 12 z 2 + 2 a 5 z -5 a 3 z -10 a z -2 z a -1 + z a -3 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11n221. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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