L11a401

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_18,12,19,11 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_22,9,11,10 X_4,17,1,18 X_10,19,5,20 X_12,6,13,5 X_14,21,15,22 X_20,13,21,14
Gauss code	{1, -4, 3, -7}, {9, -1, 5, -3, 6, -8}, {2, -9, 11, -10, 4, -5, 7, -2, 8, -11, 10, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 -2 v u 4 - v 2 w u 4 + 2 v w u 4 - w u 4 + u 4 -3 v 2 u 3 + 6 v u 3 + 3 v 2 w u 3 -6 v w u 3 + 3 w u 3 -3 u 3 + 4 v 2 u 2 -8 v u 2 -4 v 2 w u 2 + 8 v w u 2 -4 w u 2 + 4 u 2 -3 v 2 u + 6 v u + 3 v 2 w u -6 v w u + 3 w u -3 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$q 3 -5 q 2 + 12 q -19 + 28 q -1 -30 q -2 + 32 q -3 -27 q -4 + 20 q -5 -12 q -6 + 5 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + 2 a 4 z 6 -4 a 2 z 6 + z 6 - a 6 z 4 + 5 a 4 z 4 -6 a 2 z 4 + 2 z 4 - a 6 z 2 + 3 a 4 z 2 -3 a 2 z 2 + z 2 - a 2 + 1 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$4 a 4 z 10 + 4 a 2 z 10 + 13 a 5 z 9 + 24 a 3 z 9 + 11 a z 9 + 17 a 6 z 8 + 28 a 4 z 8 + 22 a 2 z 8 + 11 z 8 + 12 a 7 z 7 -6 a 5 z 7 -37 a 3 z 7 -14 a z 7 + 5 z 7 a -1 + 5 a 8 z 6 -25 a 6 z 6 -73 a 4 z 6 -67 a 2 z 6 + z 6 a -2 -23 z 6 + a 9 z 5 -14 a 7 z 5 -17 a 5 z 5 + 2 a 3 z 5 -4 a z 5 -8 z 5 a -1 -3 a 8 z 4 + 14 a 6 z 4 + 51 a 4 z 4 + 51 a 2 z 4 - z 4 a -2 + 16 z 4 + 5 a 7 z 3 + 10 a 5 z 3 + 8 a 3 z 3 + 6 a z 3 + 3 z 3 a -1 -4 a 6 z 2 -12 a 4 z 2 -12 a 2 z 2 -4 z 2 - a 3 z - a z + a 4 + a 2 + 1 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

-2 is the signature of L11a401. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -2$	${\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{17}$
$r = -1$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{17}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$