L11n168

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_18,11,19,12 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,13,18 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, -7, 8, 9, -5, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + v 2 u 3 -3 v u 3 + u 3 -2 v 2 u 2 + 3 v u 2 -2 u 2 + v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$\frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 4 z 3 a 7 + 5 z a 7 + a 7 z -1 - z 7 a 5 -6 z 5 a 5 -13 z 3 a 5 -10 z a 5 - a 5 z -1 + z 5 a 3 + 3 z 3 a 3 + 2 z a 3$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -2 z a 11 -2 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -2 z 7 a 9 + 3 z 5 a 9 + z 3 a 9 -2 z 8 a 8 + 5 z 6 a 8 -7 z 4 a 8 + 5 z 2 a 8 - z 9 a 7 + 2 z 7 a 7 -5 z 5 a 7 + 6 z 3 a 7 -5 z a 7 + a 7 z -1 -3 z 8 a 6 + 10 z 6 a 6 -18 z 4 a 6 + 10 z 2 a 6 - a 6 - z 9 a 5 + 4 z 7 a 5 -12 z 5 a 5 + 14 z 3 a 5 -9 z a 5 + a 5 z -1 - z 8 a 4 + 3 z 6 a 4 -7 z 4 a 4 + 5 z 2 a 4 -3 z 5 a 3 + 6 z 3 a 3 -2 z a 3 - z 4 a 2 + 2 z 2 a 2$ (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 L11n168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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