L11a524

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + v u 3 + 2 v 2 w u 3 -2 v w u 3 + 2 v 2 u 2 + 3 v 2 w 2 u 2 -4 v w 2 u 2 + w 2 u 2 -5 v u 2 -5 v 2 w u 2 + 9 v w u 2 -3 w u 2 + 2 u 2 - v 2 u -2 v 2 w 2 u + 5 v w 2 u -2 w 2 u + 4 v u + 3 v 2 w u -9 v w u + 5 w u -3 u - v w 2 + w 2 - v + 2 v w -2 w + 1$ (db)
Jones polynomial	$q 6 -4 q 5 + 10 q 4 -16 q 3 + 24 q 2 -26 q + 28-24 q -1 + 18 q -2 -11 q -3 + 5 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 -2 z 6 a -2 + 4 z 6 -2 a 2 z 4 -6 z 4 a -2 + z 4 a -4 + 5 z 4 -5 z 2 a -2 + 2 z 2 a -4 + z 2 + a 2 -2 a -2 + a -4 -2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$3 z 10 a -2 + 3 z 10 + 10 a z 9 + 18 z 9 a -1 + 8 z 9 a -3 + 14 a 2 z 8 + 16 z 8 a -2 + 8 z 8 a -4 + 22 z 8 + 11 a 3 z 7 -2 a z 7 -28 z 7 a -1 -11 z 7 a -3 + 4 z 7 a -5 + 5 a 4 z 6 -19 a 2 z 6 -52 z 6 a -2 -18 z 6 a -4 + z 6 a -6 -57 z 6 + a 5 z 5 -14 a 3 z 5 -20 a z 5 + 2 z 5 a -1 - z 5 a -3 -8 z 5 a -5 -4 a 4 z 4 + 6 a 2 z 4 + 50 z 4 a -2 + 16 z 4 a -4 -2 z 4 a -6 + 42 z 4 + 4 a 3 z 3 + 13 a z 3 + 11 z 3 a -1 + 6 z 3 a -3 + 4 z 3 a -5 -22 z 2 a -2 -10 z 2 a -4 + z 2 a -6 -11 z 2 -5 z a -1 -5 z a -3 - a 2 + 6 a -2 + 4 a -4 + 2 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11a524. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$