L11a520

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + 2 v w 2 u 3 - w 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -4 v w u 3 + 2 w u 3 - u 3 + 3 v 2 u 2 + 2 v 2 w 2 u 2 -5 v w 2 u 2 + 3 w 2 u 2 -5 v u 2 -5 v 2 w u 2 + 10 v w u 2 -5 w u 2 + 2 u 2 -3 v 2 u -2 v 2 w 2 u + 5 v w 2 u -3 w 2 u + 5 v u + 5 v 2 w u -10 v w u + 5 w u -2 u + v 2 + v 2 w 2 -2 v w 2 + w 2 -2 v -2 v 2 w + 4 v w -2 w + 1$ (db)
Jones polynomial	$- q 5 + 5 q 4 -13 q 3 + 23 q 2 -31 q + 37-36 q -1 + 33 q -2 -23 q -3 + 15 q -4 -6 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 4 z 6 + a 4 z 4 -4 a 2 z 4 -2 z 4 a -2 + 7 z 4 -2 a 2 z 2 -2 z 2 a -2 + 4 z 2 - a 2 + 1 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$7 a 2 z 10 + 7 z 10 + 17 a 3 z 9 + 36 a z 9 + 19 z 9 a -1 + 15 a 4 z 8 + 22 a 2 z 8 + 21 z 8 a -2 + 28 z 8 + 6 a 5 z 7 -27 a 3 z 7 -67 a z 7 -21 z 7 a -1 + 13 z 7 a -3 + a 6 z 6 -27 a 4 z 6 -72 a 2 z 6 -32 z 6 a -2 + 5 z 6 a -4 -81 z 6 -6 a 5 z 5 + 6 a 3 z 5 + 26 a z 5 -13 z 5 a -3 + z 5 a -5 + 11 a 4 z 4 + 41 a 2 z 4 + 21 z 4 a -2 -2 z 4 a -4 + 53 z 4 + a 3 z 3 - a z 3 + 3 z 3 a -1 + 5 z 3 a -3 + 2 a 4 z 2 -2 a 2 z 2 -6 z 2 a -2 -10 z 2 - a 3 z - a z + a 4 + a 2 + 1 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a520. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{20}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = -1$	${\mathbb Z}^{18}\oplus{\mathbb Z}_2^{18}$	${\mathbb Z}^{18}$
$r = 0$	${\mathbb Z}^{19}\oplus{\mathbb Z}_2^{18}$	${\mathbb Z}^{20}$
$r = 1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{17}$	${\mathbb Z}^{17}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$