L11n187

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,12,7,11 X_15,20,16,21 X_18,10,19,9 X_10,20,11,19 X_21,14,22,15 X_16,6,17,5 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, 7, -4, -8, 11, -5, 6, 4, -7, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 -2 v 2 u 3 + 4 v u 3 -2 u 3 + 3 v 2 u 2 -5 v u 2 + 3 u 2 -2 v 2 u + 4 v u -2 u + v 2 - v$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+6 q^{13/2}-9 q^{11/2}+10 q^{9/2}-11 q^{7/2}+9 q^{5/2}-7 q^{3/2}+4 \sqrt{q}-\frac{2}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2 z 5 a -3 - z 5 a -5 + 2 z 3 a -1 -6 z 3 a -3 - z 3 a -5 + z 3 a -7 + 4 z a -1 -5 z a -3 + z a -5 + z a -7 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 - z 8 a -2 -5 z 8 a -4 -4 z 8 a -6 -6 z 7 a -5 -6 z 7 a -7 + 10 z 6 a -4 + 5 z 6 a -6 -5 z 6 a -8 -3 z 5 a -1 + 17 z 5 a -5 + 11 z 5 a -7 -3 z 5 a -9 + 4 z 4 a -2 -6 z 4 a -4 -3 z 4 a -6 + 6 z 4 a -8 - z 4 a -10 + 8 z 3 a -1 + 7 z 3 a -3 -15 z 3 a -5 -11 z 3 a -7 + 3 z 3 a -9 - z 2 a -2 + 2 z 2 a -4 -2 z 2 a -8 + z 2 a -10 -5 z a -1 -5 z a -3 + 4 z a -5 + 4 z a -7 - a -2 + a -1 z -1 + a -3 z -1$ (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 =

3 is the signature of L11n187. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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