L11n185

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_21,1,22,6 X_18,8,19,7 X_3,10,4,11 X_15,12,16,13 X_5,14,6,15 X_13,4,14,5 X_11,16,12,17 X_22,18,7,17 X_2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, 11, 2, -3, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 3 v u 2 - u 2 + 2 v 2 u -5 v u + 2 u - v 2 + 3 v -2$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+4 q^{5/2}-6 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 5 - z 5 a -1 + a 3 z 3 -2 a z 3 -3 z 3 a -1 + z 3 a -3 + a 3 z -4 z a -1 + 2 z a -3 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -2 a 2 z 8 -2 z 8 a -2 -4 z 8 - a 3 z 7 + a z 7 -2 z 7 a -3 + 7 a 2 z 6 + 5 z 6 a -2 - z 6 a -4 + 13 z 6 + a 3 z 5 + 2 a z 5 + 8 z 5 a -1 + 7 z 5 a -3 -3 a 4 z 4 -13 a 2 z 4 + z 4 a -2 + 4 z 4 a -4 -13 z 4 - a 5 z 3 -3 a 3 z 3 -6 a z 3 -10 z 3 a -1 -6 z 3 a -3 + 3 a 4 z 2 + 6 a 2 z 2 -5 z 2 a -2 -4 z 2 a -4 + 2 z 2 + a 5 z + a 3 z + 4 a z + 7 z a -1 + 3 z a -3 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11n185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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