L11n391

(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_5,12,6,13 X₈₄₉₃ X_13,22,14,19 X_9,20,10,21 X_19,10,20,11 X_21,14,22,15 X_11,18,12,5 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, -5, -7, 8, -10, 4, -6, 9, 11, -2, 3, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 -2 v u 2 + 2 v w u 2 - w u 2 + u 2 + v u - v w u + 2 w u -2 u - w + 1$ (db)
Jones polynomial	$1-3 q -1 + 6 q -2 -5 q -3 + 7 q -4 -5 q -5 + 5 q -6 -2 q -7 + q -9 - q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + 2 z 2 a 8 + 4 a 8 z -2 + 6 a 8 -4 z 2 a 6 -5 a 6 z -2 -10 a 6 - z 6 a 4 -3 z 4 a 4 + 2 a 4 z -2 + 4 a 4 + z 4 a 2 + 2 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 7 a 11 -6 z 5 a 11 + 9 z 3 a 11 -5 z a 11 + a 11 z -1 + z 8 a 10 -7 z 6 a 10 + 12 z 4 a 10 -8 z 2 a 10 - a 10 z -2 + 3 a 10 + z 7 a 9 -12 z 5 a 9 + 29 z 3 a 9 -20 z a 9 + 5 a 9 z -1 + 2 z 8 a 8 -15 z 6 a 8 + 33 z 4 a 8 -25 z 2 a 8 -4 a 8 z -2 + 13 a 8 + z 9 a 7 -2 z 7 a 7 -11 z 5 a 7 + 35 z 3 a 7 -30 z a 7 + 9 a 7 z -1 + 4 z 8 a 6 -17 z 6 a 6 + 25 z 4 a 6 -23 z 2 a 6 -5 a 6 z -2 + 16 a 6 + z 9 a 5 + z 7 a 5 -14 z 5 a 5 + 19 z 3 a 5 -15 z a 5 + 5 a 5 z -1 + 3 z 8 a 4 -8 z 6 a 4 + z 4 a 4 -3 z 2 a 4 -2 a 4 z -2 + 6 a 4 + 3 z 7 a 3 -9 z 5 a 3 + 4 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 3 z 2 a 2 - a 2$ (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 =

-4 is the signature of L11n391. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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