L11a186

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_22,18,7,17 X_16,5,17,6 X_4,15,5,16 X_14,22,15,21 X_18,12,19,11 X_20,14,21,13 X_12,20,13,19 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -9, 8, -6, 5, -4, 3, -7, 9, -8, 6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 6 - v 2 u 5 + 3 v u 5 -2 u 5 + 2 v 2 u 4 -5 v u 4 + 2 u 4 -2 v 2 u 3 + 5 v u 3 -2 u 3 + 2 v 2 u 2 -5 v u 2 + 2 u 2 -2 v 2 u + 3 v u - u - v$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+6 q^{7/2}-9 q^{5/2}+12 q^{3/2}-13 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- a z 7 - z 7 a -1 + a 3 z 5 -5 a z 5 -4 z 5 a -1 + z 5 a -3 + 4 a 3 z 3 -10 a z 3 -4 z 3 a -1 + 3 z 3 a -3 + 5 a 3 z -11 a z + z a -1 + 2 z a -3 + 3 a 3 z -1 -5 a z -1 + 2 a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -6 a z 9 -3 z 9 a -1 -2 a 4 z 8 -2 a 2 z 8 -5 z 8 a -2 -5 z 8 - a 5 z 7 + 14 a 3 z 7 + 22 a z 7 + z 7 a -1 -6 z 7 a -3 + 8 a 4 z 6 + 18 a 2 z 6 + 6 z 6 a -2 -5 z 6 a -4 + 21 z 6 + 5 a 5 z 5 -26 a 3 z 5 -36 a z 5 + 7 z 5 a -1 + 9 z 5 a -3 -3 z 5 a -5 -8 a 4 z 4 -28 a 2 z 4 + 6 z 4 a -4 - z 4 a -6 -27 z 4 -7 a 5 z 3 + 28 a 3 z 3 + 40 a z 3 -5 z 3 a -1 -7 z 3 a -3 + 3 z 3 a -5 + a 4 z 2 + 20 a 2 z 2 -4 z 2 a -2 -3 z 2 a -4 + z 2 a -6 + 19 z 2 + 2 a 5 z -15 a 3 z -22 a z -3 z a -1 + 2 z a -3 -5 a 2 + a -4 -5 + 3 a 3 z -1 + 5 a z -1 + 2 a -1 z -1$ (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 L11a186. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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