L11a503

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_18,12,19,11 X_12,18,7,17 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -6}, {11, -4, 7, -9, 6, -5, 8, -7, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w 2 u 3 + v u 3 + v 2 w u 3 - v w u 3 - u 3 + 3 v 2 w 2 u 2 -2 v w 2 u 2 -2 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 + 3 u 2 -3 v 2 w 2 u + 2 v w 2 u + 2 v u + 2 v 2 w u -4 v w u + 2 w u -3 u + v 2 w 2 - v w 2 + v w - w + 1$ (db)
Jones polynomial	$- q 4 + 2 q 3 -5 q 2 + 9 q -11 + 15 q -1 -14 q -2 + 14 q -3 -10 q -4 + 7 q -5 -3 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + 2 z 6 + 4 a 4 z 4 -15 a 2 z 4 - z 4 a -2 + 10 z 4 + 6 a 4 z 2 -21 a 2 z 2 -4 z 2 a -2 + 18 z 2 + 5 a 4 -16 a 2 -4 a -2 + 15 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 4 a 3 z 9 + 6 a z 9 + 2 z 9 a -1 + 7 a 4 z 8 + 9 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + 7 a 5 z 7 - a 3 z 7 -12 a z 7 -3 z 7 a -1 + z 7 a -3 + 6 a 6 z 6 -12 a 4 z 6 -35 a 2 z 6 -8 z 6 a -2 -25 z 6 + 3 a 7 z 5 -8 a 5 z 5 -15 a 3 z 5 -11 a z 5 -12 z 5 a -1 -5 z 5 a -3 + a 8 z 4 -8 a 6 z 4 + 11 a 4 z 4 + 47 a 2 z 4 + 10 z 4 a -2 + 37 z 4 -2 a 7 z 3 + 2 a 5 z 3 + 23 a 3 z 3 + 39 a z 3 + 28 z 3 a -1 + 8 z 3 a -3 - a 8 z 2 + 6 a 6 z 2 -9 a 4 z 2 -39 a 2 z 2 -7 z 2 a -2 -30 z 2 + a 5 z -16 a 3 z -33 a z -21 z a -1 -5 z a -3 -2 a 6 + 6 a 4 + 20 a 2 + 4 a -2 + 17 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

-2 is the signature of L11a503. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -6$	${\mathbb Z}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$