L11a522

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + v u 3 + 2 v 2 w u 3 -3 v w u 3 + w u 3 + 2 v 2 u 2 + 2 v 2 w 2 u 2 -3 v w 2 u 2 + w 2 u 2 -4 v u 2 -4 v 2 w u 2 + 8 v w u 2 -3 w u 2 + u 2 - v 2 u - v 2 w 2 u + 4 v w 2 u -2 w 2 u + 3 v u + 3 v 2 w u -8 v w u + 4 w u -2 u - v w 2 + w 2 - v - v 2 w + 3 v w -2 w + 1$ (db)
Jones polynomial	$q 6 -4 q 5 + 10 q 4 -15 q 3 + 22 q 2 -24 q + 25-21 q -1 + 16 q -2 -9 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 10 z 4 -3 a 2 z 2 -9 z 2 a -2 + 2 z 2 a -4 + 8 z 2 -5 a -2 + 2 a -4 + 3-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$3 z 10 a -2 + 3 z 10 + 9 a z 9 + 17 z 9 a -1 + 8 z 9 a -3 + 11 a 2 z 8 + 13 z 8 a -2 + 8 z 8 a -4 + 16 z 8 + 8 a 3 z 7 -10 a z 7 -36 z 7 a -1 -14 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -19 a 2 z 6 -52 z 6 a -2 -19 z 6 a -4 + z 6 a -6 -55 z 6 + a 5 z 5 -11 a 3 z 5 -2 a z 5 + 21 z 5 a -1 + 3 z 5 a -3 -8 z 5 a -5 -5 a 4 z 4 + 15 a 2 z 4 + 62 z 4 a -2 + 16 z 4 a -4 -2 z 4 a -6 + 64 z 4 - a 5 z 3 + 4 a 3 z 3 + 9 a z 3 + 5 z 3 a -1 + 4 z 3 a -3 + 3 z 3 a -5 + a 4 z 2 -7 a 2 z 2 -37 z 2 a -2 -12 z 2 a -4 + z 2 a -6 -32 z 2 - a 3 z -3 a z -8 z a -1 -6 z a -3 + a 2 + 12 a -2 + 6 a -4 + 8 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11a522. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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