L11a500

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_22,14,19,13 X_20,8,21,7 X_10,20,11,19 X_16,10,17,9 X_14,18,15,17 X_8,16,9,15 X_18,22,5,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {5, -4, 9, -3}, {10, -1, 4, -8, 6, -5, 11, -2, 3, -7, 8, -6, 7, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 + 4 v u 4 -2 v w u 4 + 2 w u 4 -4 u 4 -8 v u 3 + 6 v w u 3 -6 w u 3 + 7 u 3 + 6 v u 2 -7 v w u 2 + 8 w u 2 -6 u 2 -2 v u + 4 v w u -4 w u + 2 u - v w$ (db)
Jones polynomial	$- q 8 + 5 q 7 -12 q 6 + 18 q 5 -23 q 4 + 27 q 3 -25 q 2 + 22 q -14 + 9 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 - z 4 a -2 + z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 -7 z 2 a -2 + 3 z 2 a -4 - z 2 + a 2 -10 a -2 + 7 a -4 - a -6 + 3-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 a -4 + 5 z 9 a -1 + 14 z 9 a -3 + 9 z 9 a -5 + 16 z 8 a -2 + 25 z 8 a -4 + 15 z 8 a -6 + 6 z 8 + 3 a z 7 + z 7 a -1 -11 z 7 a -3 + 3 z 7 a -5 + 12 z 7 a -7 + a 2 z 6 -47 z 6 a -2 -60 z 6 a -4 -23 z 6 a -6 + 5 z 6 a -8 -14 z 6 -6 a z 5 -24 z 5 a -1 -31 z 5 a -3 -30 z 5 a -5 -16 z 5 a -7 + z 5 a -9 -3 a 2 z 4 + 53 z 4 a -2 + 47 z 4 a -4 + 11 z 4 a -6 -3 z 4 a -8 + 17 z 4 + 3 a z 3 + 34 z 3 a -1 + 52 z 3 a -3 + 26 z 3 a -5 + 5 z 3 a -7 + 3 a 2 z 2 -40 z 2 a -2 -25 z 2 a -4 -4 z 2 a -6 -16 z 2 -20 z a -1 -33 z a -3 -15 z a -5 -2 z a -7 - a 2 + 21 a -2 + 14 a -4 + 2 a -6 + 9 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

2 is the signature of L11a500. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$