L11n406

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_11,22,12,19 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_19,12,20,13 X_14,9,15,10 X_2,14,3,13 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q 3 - q 2 + 2 q + q -1 + 2 q -4 - q -5 + q -6 - q -7$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 2 a 6 -2 a 6 + z 4 a 4 + 4 z 2 a 4 + 4 a 4 + a 2 z -2 - z 4 -4 z 2 -2 z -2 -4 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 6 z 8 + 2 a 4 z 8 + 2 a 2 z 8 + z 8 + a 7 z 7 + 2 a 5 z 7 -6 a 3 z 7 -6 a z 7 + z 7 a -1 -6 a 6 z 6 -14 a 4 z 6 -13 a 2 z 6 + z 6 a -2 -4 z 6 -6 a 7 z 5 -12 a 5 z 5 + 8 a 3 z 5 + 10 a z 5 -4 z 5 a -1 + 11 a 6 z 4 + 30 a 4 z 4 + 22 a 2 z 4 -5 z 4 a -2 -2 z 4 + 10 a 7 z 3 + 18 a 5 z 3 -2 a 3 z 3 -10 a z 3 -10 a 6 z 2 -28 a 4 z 2 -12 a 2 z 2 + 6 z 2 a -2 + 12 z 2 -4 a 7 z -8 a 5 z + 8 a z + 4 z a -1 + 4 a 6 + 8 a 4 -4 a -2 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

-1 is the signature of L11n406. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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