L11n186

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,10,19,9 X_19,22,20,7 X_13,20,14,21 X_21,14,22,15 X_10,16,11,15 X_16,6,17,5 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, -5, 6, 7, -8, 11, -3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 2 v 2 u 3 -3 v u 3 + u 3 -2 v 2 u 2 + 5 v u 2 -2 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -3 a 3 z 3 + 8 a z 3 -3 z 3 a -1 -2 a 3 z + 3 a z -2 z a -1 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -4 a 2 z 8 -3 z 8 + 2 a 3 z 7 -2 a z 7 -4 z 7 a -1 + 2 a 4 z 6 + 11 a 2 z 6 -3 z 6 a -2 + 6 z 6 -3 a 5 z 5 -4 a 3 z 5 + 10 a z 5 + 10 z 5 a -1 - z 5 a -3 - a 6 z 4 -5 a 4 z 4 -14 a 2 z 4 + 7 z 4 a -2 -3 z 4 + 5 a 5 z 3 -14 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + 2 a 6 z 2 + 4 a 4 z 2 + 4 a 2 z 2 -2 z 2 a -2 - a 5 z + 4 a z + 3 z a -1 - a 2 + a 3 z -1 + a z -1$ (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 L11n186. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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