L11n286

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_15,1,16,4 X_21,18,22,19 X_9,21,10,20 X_19,5,20,10 X_11,16,12,17 X_17,22,18,11
Gauss code	{1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v w u 2 + 2 w u 2 + 2 v 2 u -3 v u + 3 v w u -2 w u -2 v 2 + 2 v$ (db)
Jones polynomial	$2 q 2 -2 q + 5-5 q -1 + 6 q -2 -5 q -3 + 5 q -4 -3 q -5 + 2 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + z 4 a 4 + 2 z 2 a 4 + 2 a 4 + z 4 a 2 + z 2 a 2 + a 2 z -2 + a 2 -3 z 2 -2 z -2 -4 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 4 a 4 z 8 + 2 a 2 z 8 + a 7 z 7 -2 a 5 z 7 - a 3 z 7 + 2 a z 7 -10 a 6 z 6 -17 a 4 z 6 -5 a 2 z 6 + 2 z 6 -5 a 7 z 5 -5 a 5 z 5 -4 a 3 z 5 -3 a z 5 + z 5 a -1 + 15 a 6 z 4 + 23 a 4 z 4 + 3 a 2 z 4 -5 z 4 + 7 a 7 z 3 + 10 a 5 z 3 + 2 a 3 z 3 -2 a z 3 - z 3 a -1 -8 a 6 z 2 -16 a 4 z 2 - a 2 z 2 + 3 z 2 a -2 + 10 z 2 -2 a 7 z -4 a 5 z + 6 a z + 4 z a -1 + 2 a 6 + 4 a 4 -2 a 2 -4 a -2 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11n286. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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