L11n407

(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_22,12,19,11 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_12,20,13,19 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 $u$ , $v$ , $w$ , ...)	$2 v u 3 -2 v w u 3 -4 v u 2 + 4 v w u 2 -2 w u 2 + 2 u 2 + 2 v u -2 v w u + 4 w u -4 u -2 w + 2$ (db)
Jones polynomial	$3 q 5 -5 q 4 + 9 q 3 -9 q 2 + 11 q -10 + 8 q -1 -5 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 - a 2 z 4 + 3 z 4 a -2 - z 4 a -4 + 3 z 4 -2 a 2 z 2 + 3 z 2 a -2 -3 z 2 a -4 + 2 z 2 + 3 a -2 -4 a -4 + a -6 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 3 a 2 z 8 + 6 z 8 a -2 + 9 z 8 + a 3 z 7 -3 a z 7 + 4 z 7 a -1 + 8 z 7 a -3 -13 a 2 z 6 -12 z 6 a -2 + 7 z 6 a -4 -32 z 6 -4 a 3 z 5 -10 a z 5 -25 z 5 a -1 -16 z 5 a -3 + 3 z 5 a -5 + 17 a 2 z 4 + 4 z 4 a -2 -12 z 4 a -4 + 33 z 4 + 5 a 3 z 3 + 18 a z 3 + 20 z 3 a -1 + 7 z 3 a -3 -6 a 2 z 2 -3 z 2 a -2 + 11 z 2 a -4 + 6 z 2 a -6 -14 z 2 -2 a 3 z -6 a z -6 z a -1 + 2 z a -3 + 4 z a -5 + a 2 - a -2 -6 a -4 -4 a -6 + 3-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L11n407. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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