L11a176

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 3 v 2 u 5 -4 v u 5 + u 5 -5 v 2 u 4 + 8 v u 4 -3 u 4 + 5 v 2 u 3 -11 v u 3 + 5 u 3 -3 v 2 u 2 + 8 v u 2 -5 u 2 + v 2 u -4 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-24 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 9 a -1 - a z 7 + 6 z 7 a -1 - z 7 a -3 -4 a z 5 + 13 z 5 a -1 -4 z 5 a -3 -5 a z 3 + 12 z 3 a -1 -5 z 3 a -3 -2 a z + 5 z a -1 -2 z a -3 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 -7 a z 9 -15 z 9 a -1 -8 z 9 a -3 -7 a 2 z 8 -10 z 8 a -2 -10 z 8 a -4 -7 z 8 -4 a 3 z 7 + 12 a z 7 + 32 z 7 a -1 + 8 z 7 a -3 -8 z 7 a -5 - a 4 z 6 + 15 a 2 z 6 + 27 z 6 a -2 + 15 z 6 a -4 -4 z 6 a -6 + 24 z 6 + 9 a 3 z 5 -6 a z 5 -29 z 5 a -1 - z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -8 a 2 z 4 -19 z 4 a -2 -8 z 4 a -4 + 5 z 4 a -6 -16 z 4 -5 a 3 z 3 + 6 a z 3 + 18 z 3 a -1 -6 z 3 a -5 + z 3 a -7 - a 4 z 2 + a 2 z 2 + 4 z 2 a -2 + z 2 a -4 - z 2 a -6 + 4 z 2 -3 a z -6 z a -1 -2 z a -3 + z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11a176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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