L11a504

(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_22,17,13,18 X_20,15,21,16 X_16,21,17,22 X_12,19,7,20 X_18,11,19,12 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, {11, -4, 6, -7, 5, -9, 8, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 w 2 u 3 + v w 2 u 3 + v u 3 + v 2 w u 3 -2 v w u 3 + w u 3 - u 3 + 2 v 2 w 2 u 2 - v w 2 u 2 - v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + 2 u 2 -2 v 2 w 2 u + v w 2 u + v u + v 2 w u -2 v w u + w u -2 u + v 2 w 2 - v w 2 - v - v 2 w + 2 v w - w + 2$ (db)
Jones polynomial	$q -4 - q -5 + 5 q -6 -6 q -7 + 10 q -8 -11 q -9 + 12 q -10 -11 q -11 + 9 q -12 -6 q -13 + 3 q -14 - q -15$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$- a 14 z -2 - a 14 - z 6 a 12 -3 z 4 a 12 + 2 z 2 a 12 + 4 a 12 z -2 + 9 a 12 + z 8 a 10 + 4 z 6 a 10 - z 4 a 10 -18 z 2 a 10 -5 a 10 z -2 -19 a 10 + z 8 a 8 + 7 z 6 a 8 + 18 z 4 a 8 + 21 z 2 a 8 + 2 a 8 z -2 + 11 a 8$ (db)
Kauffman polynomial	$z 3 a 19 + 3 z 4 a 18 + 6 z 5 a 17 -4 z 3 a 17 + z a 17 + 9 z 6 a 16 -13 z 4 a 16 + 5 z 2 a 16 - a 16 + 10 z 7 a 15 -20 z 5 a 15 + 9 z 3 a 15 -2 z a 15 + a 15 z -1 + 8 z 8 a 14 -17 z 6 a 14 + 3 z 4 a 14 + 3 z 2 a 14 - a 14 z -2 + 4 z 9 a 13 -3 z 7 a 13 -23 z 5 a 13 + 33 z 3 a 13 -19 z a 13 + 5 a 13 z -1 + z 10 a 12 + 6 z 8 a 12 -32 z 6 a 12 + 37 z 4 a 12 -20 z 2 a 12 -4 a 12 z -2 + 13 a 12 + 5 z 9 a 11 -16 z 7 a 11 - z 5 a 11 + 39 z 3 a 11 -35 z a 11 + 9 a 11 z -1 + z 10 a 10 - z 8 a 10 -13 z 6 a 10 + 36 z 4 a 10 -39 z 2 a 10 -5 a 10 z -2 + 22 a 10 + z 9 a 9 -3 z 7 a 9 -4 z 5 a 9 + 20 z 3 a 9 -19 z a 9 + 5 a 9 z -1 + z 8 a 8 -7 z 6 a 8 + 18 z 4 a 8 -21 z 2 a 8 -2 a 8 z -2 + 11 a 8$ (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 =

-8 is the signature of L11a504. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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