L11a166

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 5 v 2 u 3 -5 v u 3 + u 3 -5 v 2 u 2 + 7 v u 2 -5 u 2 + v 2 u -5 v u + 5 u + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-5 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + 2 z 5 a 5 + 6 z 3 a 5 + 4 z a 5 + a 5 z -1 - z 7 a 3 -4 z 5 a 3 -6 z 3 a 3 -6 z a 3 -2 a 3 z -1 + 2 z 5 a + 7 z 3 a + 6 z a + 2 a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 3 z 3 a 9 - z a 9 -5 z 6 a 8 + 5 z 4 a 8 -2 z 2 a 8 -6 z 7 a 7 + 7 z 5 a 7 -4 z 3 a 7 + z a 7 -5 z 8 a 6 + 4 z 6 a 6 + 2 z 4 a 6 -2 z 2 a 6 -3 z 9 a 5 - z 7 a 5 + 12 z 5 a 5 -12 z 3 a 5 + 6 z a 5 - a 5 z -1 - z 10 a 4 -5 z 8 a 4 + 15 z 6 a 4 -8 z 4 a 4 + z 2 a 4 -5 z 9 a 3 + 10 z 7 a 3 + 4 z 5 a 3 -15 z 3 a 3 + 10 z a 3 -2 a 3 z -1 - z 10 a 2 -2 z 8 a 2 + 14 z 6 a 2 -13 z 4 a 2 + 3 z 2 a 2 - a 2 -2 z 9 a + 4 z 7 a + 7 z 5 a -18 z 3 a + 11 z a -2 a z -1 -2 z 8 + 8 z 6 -9 z 4 + 3 z 2 - z 7 a -1 + 5 z 5 a -1 -8 z 3 a -1 + 5 z a -1 - a -1 z -1$ (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 L11a166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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