L11a507

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 -2 v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + u 3 -2 v 2 u 2 - v 2 w 2 u 2 + 2 v w 2 u 2 - w 2 u 2 + 4 v u 2 + 3 v 2 w u 2 -7 v w u 2 + 3 w u 2 -2 u 2 + v 2 u + 2 v 2 w 2 u -4 v w 2 u + 2 w 2 u -2 v u -3 v 2 w u + 7 v w u -3 w u + u - v 2 w 2 + 2 v w 2 - w 2 + v 2 w -2 v w + w$ (db)
Jones polynomial	$q 9 -4 q 8 + 8 q 7 -13 q 6 + 19 q 5 -20 q 4 + 22 q 3 -18 q 2 + 14 q -8 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + z 4 a -4 -2 z 4 a -6 - z 4 + 3 z 2 a -2 -2 z 2 a -6 + z 2 a -8 - z 2 + 2 a -2 -2 a -4 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 6 z 9 a -3 + 12 z 9 a -5 + 6 z 9 a -7 + 8 z 8 a -2 + 12 z 8 a -4 + 11 z 8 a -6 + 7 z 8 a -8 + 7 z 7 a -1 -3 z 7 a -3 -24 z 7 a -5 -10 z 7 a -7 + 4 z 7 a -9 -10 z 6 a -2 -35 z 6 a -4 -41 z 6 a -6 -19 z 6 a -8 + z 6 a -10 + 4 z 6 + a z 5 -10 z 5 a -1 -6 z 5 a -3 + 14 z 5 a -5 - z 5 a -7 -10 z 5 a -9 + 2 z 4 a -2 + 34 z 4 a -4 + 44 z 4 a -6 + 16 z 4 a -8 -2 z 4 a -10 -6 z 4 - a z 3 + 3 z 3 a -1 + 4 z 3 a -3 + 5 z 3 a -7 + 5 z 3 a -9 + z 2 a -2 -10 z 2 a -4 -15 z 2 a -6 -6 z 2 a -8 + 2 z 2 + 2 z a -3 + 2 z a -5 -2 a -2 -3 a -4 -2 a -6 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L11a507. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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