L11n208

(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_21,6,22,7 X_7,17,8,16 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 $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 - v 2 u 2 - v u + v -1$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a z 7 + a 3 z 5 -7 a z 5 + z 5 a -1 + 5 a 3 z 3 -16 a z 3 + 5 z 3 a -1 + 6 a 3 z -14 a z + 6 z a -1 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 - z 8 a -2 -2 z 8 + 7 a z 7 + 7 z 7 a -1 + 6 a 2 z 6 + 7 z 6 a -2 + 13 z 6 -2 a 3 z 5 -18 a z 5 -16 z 5 a -1 - a 4 z 4 -12 a 2 z 4 -15 z 4 a -2 -26 z 4 - a 5 z 3 + 8 a 3 z 3 + 24 a z 3 + 15 z 3 a -1 - a 6 z 2 + 2 a 4 z 2 + 11 a 2 z 2 + 10 z 2 a -2 + 18 z 2 - a 7 z + a 5 z -7 a 3 z -16 a z -7 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 =

-3 is the signature of L11n208. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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