L11a529

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - w 2 u 3 + 2 v u 3 -2 v w u 3 + 2 w u 3 - u 3 + 3 v 2 u 2 -3 v w 2 u 2 + 3 w 2 u 2 -5 v u 2 -3 v 2 w u 2 + 7 v w u 2 -5 w u 2 + 2 u 2 -3 v 2 u -2 v 2 w 2 u + 5 v w 2 u -3 w 2 u + 3 v u + 5 v 2 w u -7 v w u + 3 w u + v 2 + v 2 w 2 -2 v w 2 + w 2 -2 v 2 w + 2 v w$ (db)
Jones polynomial	$q -2 -4 q -3 + 10 q -4 -16 q -5 + 23 q -6 -25 q -7 + 26 q -8 -22 q -9 + 17 q -10 -10 q -11 + 5 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 12 + a 12 z -2 + 2 a 12 + z 4 a 10 -6 z 2 a 10 -2 a 10 z -2 -9 a 10 + 6 z 4 a 8 + 11 z 2 a 8 + a 8 z -2 + 7 a 8 + 4 z 4 a 6 + 4 z 2 a 6 + z 4 a 4$ (db)
Kauffman polynomial	$z 7 a 15 -2 z 5 a 15 + z 3 a 15 + 5 z 8 a 14 -15 z 6 a 14 + 13 z 4 a 14 - z 2 a 14 -2 a 14 + 8 z 9 a 13 -23 z 7 a 13 + 17 z 5 a 13 - z 3 a 13 + 4 z 10 a 12 + 6 z 8 a 12 -46 z 6 a 12 + 46 z 4 a 12 -11 z 2 a 12 - a 12 z -2 + 3 a 12 + 21 z 9 a 11 -53 z 7 a 11 + 26 z 5 a 11 + 9 z 3 a 11 -9 z a 11 + 2 a 11 z -1 + 4 z 10 a 10 + 20 z 8 a 10 -76 z 6 a 10 + 70 z 4 a 10 -33 z 2 a 10 -2 a 10 z -2 + 11 a 10 + 13 z 9 a 9 -13 z 7 a 9 -17 z 5 a 9 + 19 z 3 a 9 -9 z a 9 + 2 a 9 z -1 + 19 z 8 a 8 -35 z 6 a 8 + 28 z 4 a 8 -19 z 2 a 8 - a 8 z -2 + 7 a 8 + 16 z 7 a 7 -20 z 5 a 7 + 8 z 3 a 7 + 10 z 6 a 6 -8 z 4 a 6 + 4 z 2 a 6 + 4 z 5 a 5 + z 4 a 4$ (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 L11a529. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -8$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -7$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -6$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = -5$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$