L11a530

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,14,21,13 X_18,10,19,9 X_10,22,11,21 X_14,20,7,19 X_16,5,17,6 X_22,18,15,17 X₂₇₃₈ X_4,11,5,12 X_6,15,1,16
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6}, {11, -7, 8, -4, 6, -3, 5, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + v w u 3 - w u 3 + u 3 -3 v 2 u 2 - v 2 w 2 u 2 + 3 v w 2 u 2 -2 w 2 u 2 + 5 v u 2 + 3 v 2 w u 2 -8 v w u 2 + 4 w u 2 -2 u 2 + 2 v 2 u + 2 v 2 w 2 u -5 v w 2 u + 3 w 2 u -3 v u -4 v 2 w u + 8 v w u -3 w u + u - v 2 w 2 + 2 v w 2 + v 2 w - v w$ (db)
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 16 q 2 -20 q + 24-23 q -1 + 20 q -2 -14 q -3 + 9 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -3 z 2 a 4 + a 4 z -2 + 3 z 4 a 2 + z 2 a 2 -2 a 2 z -2 -3 a 2 - z 6 - z 4 -2 z 2 + z -2 + 1 + 2 z 4 a -2 + z 2 a -2 + a -2 - z 2 a -4$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 5 a 3 z 9 + 12 a z 9 + 7 z 9 a -1 + 6 a 4 z 8 + 12 a 2 z 8 + 10 z 8 a -2 + 16 z 8 + 3 a 5 z 7 - a 3 z 7 -13 a z 7 - z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -14 a 4 z 6 -34 a 2 z 6 -14 z 6 a -2 + 4 z 6 a -4 -37 z 6 -6 a 5 z 5 -18 a 3 z 5 -11 a z 5 -11 z 5 a -1 -11 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 15 a 4 z 4 + 32 a 2 z 4 + 8 z 4 a -2 -5 z 4 a -4 + 27 z 4 + 3 a 5 z 3 + 21 a 3 z 3 + 18 a z 3 + 6 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + 3 a 6 z 2 -12 a 4 z 2 -23 a 2 z 2 -2 z 2 a -2 + 2 z 2 a -4 -12 z 2 -9 a 3 z -9 a z - a 6 + 5 a 4 + 11 a 2 - a -2 + 5 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11a530. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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