L11a427

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,7,21,8 X_8,19,9,20 X_16,12,17,11 X_10,13,5,14 X_22,18,11,17 X_18,22,19,21 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -3 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 -4 v 2 u + 7 v u + 3 v 2 w u -7 v w u + 4 w u -3 u + 2 v 2 -4 v + 3 v w -2 w + 2$ (db)
Jones polynomial	$- q 4 + 4 q 3 -7 q 2 + 11 q -15 + 17 q -1 -16 q -2 + 16 q -3 -10 q -4 + 7 q -5 -3 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 6 + a 6 z -2 + a 6 -2 z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -4 a 4 + z 6 a 2 + 2 z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 4 a 2 + z 6 + 2 z 4 -2- z 4 a -2 - z 2 a -2 + a -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 5 a 3 z 9 + 10 a z 9 + 5 z 9 a -1 + 7 a 4 z 8 + 6 a 2 z 8 + 4 z 8 a -2 + 3 z 8 + 7 a 5 z 7 -2 a 3 z 7 -27 a z 7 -17 z 7 a -1 + z 7 a -3 + 6 a 6 z 6 -6 a 4 z 6 -22 a 2 z 6 -15 z 6 a -2 -25 z 6 + 3 a 7 z 5 -6 a 5 z 5 -12 a 3 z 5 + 15 a z 5 + 15 z 5 a -1 -3 z 5 a -3 + a 8 z 4 -8 a 6 z 4 -7 a 4 z 4 + 9 a 2 z 4 + 15 z 4 a -2 + 22 z 4 -2 a 7 z 3 - a 5 z 3 + 9 a 3 z 3 + 4 a z 3 -2 z 3 a -1 + 2 z 3 a -3 - a 8 z 2 + 8 a 6 z 2 + 12 a 4 z 2 + 5 a 2 z 2 -2 z 2 a -2 + 4 a 5 z -6 a z -2 z a -1 -4 a 6 -6 a 4 -3 a 2 - a -2 -1-2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

-2 is the signature of L11a427. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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