L11n409

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_22,17,19,18 X_20,12,21,11 X_10,20,11,19 X_18,21,5,22 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, -3, 9, -8, -6, 5, -2, 11, 8, -9, 3, 4, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -4 v u 2 + 4 v w u 2 -4 w u 2 + 4 u 2 + 4 v u -4 v w u + 4 w u -4 u - v + v w - w + 1$ (db)
Jones polynomial	$q 4 -4 q 3 + 9 q 2 -11 q + 14-13 q -1 + 13 q -2 -8 q -3 + 5 q -4 -2 q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + 3 a 2 z 4 + z 4 a -2 -3 z 4 -2 a 4 z 2 + 7 a 2 z 2 + z 2 a -2 -6 z 2 -2 a 4 + 6 a 2 + 2 a -2 -6 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 6 a 2 z 8 + 5 z 8 + 3 a 3 z 7 + 13 a z 7 + 10 z 7 a -1 + 2 a 4 z 6 - a 2 z 6 + 9 z 6 a -2 + 6 z 6 + 3 a 5 z 5 -5 a 3 z 5 -25 a z 5 -13 z 5 a -1 + 4 z 5 a -3 -9 a 4 z 4 -25 a 2 z 4 -11 z 4 a -2 + z 4 a -4 -28 z 4 -6 a 5 z 3 -5 a 3 z 3 + 5 a z 3 + 3 z 3 a -1 - z 3 a -3 + 10 a 4 z 2 + 30 a 2 z 2 + 6 z 2 a -2 + 26 z 2 + 2 a 5 z + 6 a 3 z + 6 a z + 2 z a -1 -4 a 4 -12 a 2 -4 a -2 -11-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11n409. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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