L11n404

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 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_2,14,3,13
Gauss code	{1, -11, 7, -5}, {-10, 8, -9, 6}, {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q 2 - q + 2 + 2 q -2 + q -3 - q -4 + q -5 - q -6 + q -7 - q -8$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -4 z 2 a 6 - a 6 z -2 -4 a 6 + z 6 a 4 + 7 z 4 a 4 + 16 z 2 a 4 + 4 a 4 z -2 + 13 a 4 - z 6 a 2 -7 z 4 a 2 -16 z 2 a 2 -5 a 2 z -2 -14 a 2 + z 4 + 4 z 2 + 2 z -2 + 5$ (db)
Kauffman polynomial	$z 5 a 9 -4 z 3 a 9 + 2 z a 9 + z 6 a 8 -4 z 4 a 8 + 2 z 2 a 8 + z 7 a 7 -5 z 5 a 7 + 6 z 3 a 7 -4 z a 7 + a 7 z -1 + z 8 a 6 -6 z 6 a 6 + 10 z 4 a 6 -10 z 2 a 6 - a 6 z -2 + 7 a 6 + 3 z 7 a 5 -21 z 5 a 5 + 40 z 3 a 5 -25 z a 5 + 5 a 5 z -1 + 3 z 8 a 4 -22 z 6 a 4 + 50 z 4 a 4 -49 z 2 a 4 -4 a 4 z -2 + 22 a 4 + z 9 a 3 -4 z 7 a 3 -7 z 5 a 3 + 34 z 3 a 3 -31 z a 3 + 9 a 3 z -1 + 3 z 8 a 2 -22 z 6 a 2 + 52 z 4 a 2 -53 z 2 a 2 -5 a 2 z -2 + 23 a 2 + z 9 a -6 z 7 a + 8 z 5 a + 4 z 3 a -12 z a + 5 a z -1 + z 8 -7 z 6 + 16 z 4 -16 z 2 -2 z -2 + 9$ (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 =

-3 is the signature of L11n404. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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