L11a169

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,11,21,12 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_18,13,19,14 X_22,16,7,15 X_12,19,13,20 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -10, 8, -5, 9, -7, 4, -8, 10, -2, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 6 v 2 u 3 -9 v u 3 + 2 u 3 -6 v 2 u 2 + 15 v u 2 -6 u 2 + 2 v 2 u -9 v u + 6 u + 2 v -2$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{19}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 3 a 3 z 5 + 3 a z 5 - z 5 a -1 -2 a 5 z 3 + 2 a 3 z 3 + 3 a z 3 -2 z 3 a -1 -2 a 3 z + 2 a z - z a -1 + a 5 z -1 -2 a 3 z -1 + 2 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 a 4 z 10 -3 a 2 z 10 -7 a 5 z 9 -15 a 3 z 9 -8 a z 9 -7 a 6 z 8 -6 a 4 z 8 -9 a 2 z 8 -10 z 8 -4 a 7 z 7 + 15 a 5 z 7 + 38 a 3 z 7 + 11 a z 7 -8 z 7 a -1 - a 8 z 6 + 18 a 6 z 6 + 31 a 4 z 6 + 34 a 2 z 6 -4 z 6 a -2 + 18 z 6 + 10 a 7 z 5 -8 a 5 z 5 -35 a 3 z 5 -3 a z 5 + 13 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -13 a 6 z 4 -35 a 4 z 4 -38 a 2 z 4 + 5 z 4 a -2 -13 z 4 -5 a 7 z 3 + 6 a 3 z 3 -6 a z 3 -6 z 3 a -1 + z 3 a -3 + 4 a 6 z 2 + 13 a 4 z 2 + 14 a 2 z 2 + 5 z 2 + 2 a 5 z + 6 a 3 z + 6 a z + 2 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11a169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$