L11a521

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + 2 v 2 w u 3 -2 v w u 3 + 2 v 2 u 2 + 2 v 2 w 2 u 2 -3 v w 2 u 2 + w 2 u 2 -2 v u 2 -4 v 2 w u 2 + 5 v w u 2 -2 w u 2 - v 2 u + 2 v w 2 u -2 w 2 u + 3 v u + 2 v 2 w u -5 v w u + 4 w u -2 u + w 2 - v + 2 v w -2 w + 1$ (db)
Jones polynomial	$q 10 -3 q 9 + 7 q 8 -11 q 7 + 16 q 6 -17 q 5 + 18 q 4 -15 q 3 + 12 q 2 -7 q + 4- q -1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 - z 6 a -2 + 5 z 6 a -4 -2 z 6 a -6 -3 z 4 a -2 + 9 z 4 a -4 -8 z 4 a -6 + z 4 a -8 - z 2 a -2 + 8 z 2 a -4 -10 z 2 a -6 + 3 z 2 a -8 + a -2 + 3 a -4 -6 a -6 + 2 a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 5 z 9 a -3 + 11 z 9 a -5 + 6 z 9 a -7 + 4 z 8 a -2 + 5 z 8 a -4 + 10 z 8 a -6 + 9 z 8 a -8 + z 7 a -1 -16 z 7 a -3 -30 z 7 a -5 -5 z 7 a -7 + 8 z 7 a -9 -15 z 6 a -2 -35 z 6 a -4 -43 z 6 a -6 -17 z 6 a -8 + 6 z 6 a -10 -3 z 5 a -1 + 12 z 5 a -3 + 16 z 5 a -5 -12 z 5 a -7 -10 z 5 a -9 + 3 z 5 a -11 + 16 z 4 a -2 + 41 z 4 a -4 + 50 z 4 a -6 + 17 z 4 a -8 -7 z 4 a -10 + z 4 a -12 + 2 z 3 a -1 - z 3 a -3 + 7 z 3 a -5 + 17 z 3 a -7 + 5 z 3 a -9 -2 z 3 a -11 -4 z 2 a -2 -17 z 2 a -4 -31 z 2 a -6 -12 z 2 a -8 + 5 z 2 a -10 - z 2 a -12 -9 z a -5 -9 z a -7 - a -2 + 5 a -4 + 11 a -6 + 5 a -8 - a -10 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 =

4 is the signature of L11a521. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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