L11a409

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_8,18,9,17 X_14,8,15,7 X_18,10,19,9 X_10,12,5,11 X_22,19,11,20 X_20,15,21,16 X_16,21,17,22 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, {6, -2, 11, -4, 8, -9, 3, -5, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v 2 w u 4 + v w u 4 -3 v 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -4 v w u 3 + 2 w u 3 + 4 v 2 u 2 -5 v u 2 -2 v 2 w u 2 + 5 v w u 2 -4 w u 2 + 2 u 2 -2 v 2 u + 4 v u -2 v w u + 3 w u -2 u - v - w + 1$ (db)
Jones polynomial	$q 7 -3 q 6 + 7 q 5 -11 q 4 + 16 q 3 -16 q 2 + 18 q -14 + 11 q -1 -7 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -6 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -15 z 4 a -2 + 4 z 4 a -4 + 9 z 4 -3 a 2 z 2 -19 z 2 a -2 + 6 z 2 a -4 + 15 z 2 -3 a 2 -13 a -2 + 4 a -4 + 12- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 + 3 a z 9 + 8 z 9 a -1 + 5 z 9 a -3 + 3 a 2 z 8 + 16 z 8 a -2 + 9 z 8 a -4 + 10 z 8 + a 3 z 7 -4 a z 7 -12 z 7 a -1 + z 7 a -3 + 8 z 7 a -5 -11 a 2 z 6 -58 z 6 a -2 -18 z 6 a -4 + 6 z 6 a -6 -45 z 6 -4 a 3 z 5 -15 a z 5 -24 z 5 a -1 -26 z 5 a -3 -10 z 5 a -5 + 3 z 5 a -7 + 13 a 2 z 4 + 72 z 4 a -2 + 21 z 4 a -4 -7 z 4 a -6 + z 4 a -8 + 56 z 4 + 6 a 3 z 3 + 30 a z 3 + 50 z 3 a -1 + 33 z 3 a -3 + 5 z 3 a -5 -2 z 3 a -7 -7 a 2 z 2 -50 z 2 a -2 -17 z 2 a -4 + 5 z 2 a -6 - z 2 a -8 -34 z 2 -4 a 3 z -19 a z -33 z a -1 -18 z a -3 + 3 a 2 + 20 a -2 + 8 a -4 - a -6 + 15 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 L11a409. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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