L11n220

(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_3,12,4,13 X_15,22,16,9 X_2,17,3,18 X_21,4,22,5 X_14,5,15,6 X_20,13,21,14 X_11,16,12,17 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 5 - v 3 u 4 + v 2 u 4 - v 2 u 3 - v u 2 + v u - u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + z 5 a 11 + 6 z 3 a 11 + 9 z a 11 + 3 a 11 z -1 - z 7 a 9 -6 z 5 a 9 -11 z 3 a 9 -8 z a 9 -2 a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -4 z a 7$ (db)
Kauffman polynomial	$- z a 15 - z 2 a 14 - a 14 - z 5 a 13 + 3 z 3 a 13 -3 z a 13 + a 13 z -1 - z 8 a 12 + 6 z 6 a 12 -13 z 4 a 12 + 12 z 2 a 12 -3 a 12 - z 9 a 11 + 7 z 7 a 11 -20 z 5 a 11 + 29 z 3 a 11 -16 z a 11 + 3 a 11 z -1 -2 z 8 a 10 + 11 z 6 a 10 -19 z 4 a 10 + 14 z 2 a 10 -3 a 10 - z 9 a 9 + 6 z 7 a 9 -13 z 5 a 9 + 16 z 3 a 9 -10 z a 9 + 2 a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -6 z 4 a 8 + z 2 a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 4 z a 7$ (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 L11n220. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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