L11n210

(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_4,19,5,20 X_21,6,22,7 X_16,7,17,8 X_8,9,1,10 X_18,14,19,13 X_15,21,16,20
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 $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v 2 u 4 - v u 4 - v 2 u 3 + v u 3 - u 3 - v 3 u 2 + v 2 u 2 - v u 2 - v 2 u + v u - v$ (db)
Jones polynomial	$\frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 5 z 3 a 7 + 7 z a 7 + 2 a 7 z -1 - z 7 a 5 -7 z 5 a 5 -17 z 3 a 5 -15 z a 5 -3 a 5 z -1 + z 5 a 3 + 4 z 3 a 3 + 4 z a 3 + a 3 z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + 4 z 3 a 11 -3 z a 11 - z 6 a 10 + 3 z 4 a 10 - z 2 a 10 - z 7 a 9 + 3 z 5 a 9 -2 z 3 a 9 + z a 9 - z 8 a 8 + 4 z 6 a 8 -6 z 4 a 8 + 3 z 2 a 8 - z 9 a 7 + 6 z 7 a 7 -15 z 5 a 7 + 16 z 3 a 7 -10 z a 7 + 2 a 7 z -1 -2 z 8 a 6 + 11 z 6 a 6 -23 z 4 a 6 + 15 z 2 a 6 -3 a 6 - z 9 a 5 + 7 z 7 a 5 -21 z 5 a 5 + 28 z 3 a 5 -18 z a 5 + 3 a 5 z -1 - z 8 a 4 + 6 z 6 a 4 -15 z 4 a 4 + 14 z 2 a 4 -3 a 4 -2 z 5 a 3 + 6 z 3 a 3 -4 z a 3 + a 3 z -1 - z 4 a 2 + 3 z 2 a 2 - a 2$ (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 L11n210. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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