L11a178

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_16,7,17,8 X_20,10,21,9 X_22,15,7,16 X_14,6,15,5 X_18,14,19,13 X_12,20,13,19 X_4,12,5,11 X_6,18,1,17 X_2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 3 v u 4 - u 4 + 5 v 2 u 3 -10 v u 3 + 4 u 3 -6 v 2 u 2 + 15 v u 2 -6 u 2 + 4 v 2 u -10 v u + 5 u - v 2 + 3 v -2$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+9 q^{11/2}-16 q^{9/2}+21 q^{7/2}-25 q^{5/2}+25 q^{3/2}-22 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -3 z 5 a -1 -3 z 5 a -3 + z 5 a -5 + 2 a z 3 -4 z 3 a -1 -3 z 3 a -3 + 2 z 3 a -5 + 2 a z -3 z a -1 + z a -3 + z a -5 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 a -4 -9 z 9 a -1 -16 z 9 a -3 -7 z 9 a -5 -15 z 8 a -2 -10 z 8 a -4 -7 z 8 a -6 -12 z 8 -9 a z 7 + 7 z 7 a -1 + 29 z 7 a -3 + 9 z 7 a -5 -4 z 7 a -7 -4 a 2 z 6 + 42 z 6 a -2 + 32 z 6 a -4 + 14 z 6 a -6 - z 6 a -8 + 21 z 6 - a 3 z 5 + 14 a z 5 + 9 z 5 a -1 -12 z 5 a -3 + 3 z 5 a -5 + 9 z 5 a -7 + 4 a 2 z 4 -34 z 4 a -2 -23 z 4 a -4 -7 z 4 a -6 + 2 z 4 a -8 -16 z 4 + a 3 z 3 -9 a z 3 -16 z 3 a -1 -5 z 3 a -3 -5 z 3 a -5 -6 z 3 a -7 + 9 z 2 a -2 + 6 z 2 a -4 + z 2 a -6 - z 2 a -8 + 5 z 2 + 4 a z + 9 z a -1 + 7 z a -3 + 3 z a -5 + z a -7 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$