L11n285

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_4,15,1,16 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 $u$ , $v$ , $w$ , ...)	$-2 v 2 w u 2 + 2 v w u 2 + v u + 2 v 2 w u - v w u -2 u -2 v + 2$ (db)
Jones polynomial	$1- q -1 + 3 q -2 -3 q -3 + 5 q -4 -4 q -5 + 5 q -6 -3 q -7 + 2 q -8 - q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 + z 4 a 8 + 4 z 2 a 8 + 2 a 8 - z 6 a 6 -4 z 4 a 6 -3 z 2 a 6 + a 6 z -2 - z 6 a 4 -4 z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -4 a 4 + z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z 3 a 11 -2 z a 11 + 2 z 4 a 10 -4 z 2 a 10 + 2 a 10 + z 7 a 9 -4 z 5 a 9 + 8 z 3 a 9 -4 z a 9 + 2 z 8 a 8 -11 z 6 a 8 + 23 z 4 a 8 -16 z 2 a 8 + 4 a 8 + z 9 a 7 -4 z 7 a 7 + 5 z 5 a 7 + 3 z 8 a 6 -14 z 6 a 6 + 21 z 4 a 6 -9 z 2 a 6 + a 6 z -2 -2 a 6 + z 9 a 5 -4 z 7 a 5 + 6 z 5 a 5 -8 z 3 a 5 + 6 z a 5 -2 a 5 z -1 + z 8 a 4 -2 z 6 a 4 -5 z 4 a 4 + 10 z 2 a 4 + 2 a 4 z -2 -7 a 4 + z 7 a 3 -3 z 5 a 3 - z 3 a 3 + 4 z a 3 -2 a 3 z -1 + z 6 a 2 -5 z 4 a 2 + 7 z 2 a 2 + a 2 z -2 -4 a 2$ (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 =

-4 is the signature of L11n285. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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