L11a509

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,10,21,9 X_14,5,15,6 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_10,4,11,3 X_18,11,19,12 X_6,15,1,16 X_4,20,5,19 X_2,21,3,22
Gauss code	{1, -11, 7, -10, 3, -9}, {5, -1, 2, -7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, -2, 11, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -3 v w u 3 + w u 3 - u 3 + 2 v 2 u 2 + 3 v 2 w 2 u 2 -4 v w 2 u 2 + w 2 u 2 -5 v u 2 -5 v 2 w u 2 + 10 v w u 2 -4 w u 2 + 3 u 2 - v 2 u -3 v 2 w 2 u + 5 v w 2 u -2 w 2 u + 4 v u + 4 v 2 w u -10 v w u + 5 w u -3 u + v 2 w 2 -2 v w 2 + w 2 - v - v 2 w + 3 v w -2 w + 1$ (db)
Jones polynomial	$q 6 -5 q 5 + 12 q 4 -20 q 3 + 28 q 2 -31 q + 33-27 q -1 + 21 q -2 -12 q -3 + 5 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 -2 z 6 a -2 + 4 z 6 -2 a 2 z 4 -5 z 4 a -2 + z 4 a -4 + 6 z 4 - a 2 z 2 -3 z 2 a -2 + z 2 a -4 + 2 z 2 + a 2 + a -2 -2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$5 z 10 a -2 + 5 z 10 + 15 a z 9 + 27 z 9 a -1 + 12 z 9 a -3 + 18 a 2 z 8 + 18 z 8 a -2 + 11 z 8 a -4 + 25 z 8 + 12 a 3 z 7 -15 a z 7 -50 z 7 a -1 -18 z 7 a -3 + 5 z 7 a -5 + 5 a 4 z 6 -28 a 2 z 6 -62 z 6 a -2 -22 z 6 a -4 + z 6 a -6 -72 z 6 + a 5 z 5 -13 a 3 z 5 -4 a z 5 + 19 z 5 a -1 + z 5 a -3 -8 z 5 a -5 -3 a 4 z 4 + 16 a 2 z 4 + 49 z 4 a -2 + 14 z 4 a -4 - z 4 a -6 + 53 z 4 + 3 a 3 z 3 + 4 a z 3 + 2 z 3 a -1 + 4 z 3 a -3 + 3 z 3 a -5 -3 a 2 z 2 -11 z 2 a -2 -4 z 2 a -4 -10 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11a509. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{18}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{17}$
$r = 1$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16}$	${\mathbb Z}^{16}$
$r = 2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{16}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$