L11n381

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$q + 2 q -1 + q -2 + q -3 - q -5 + q -6 - q -7$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + z 4 a 4 + 5 z 2 a 4 + 4 a 4 z -2 + 7 a 4 - z 4 a 2 -5 z 2 a 2 -5 a 2 z -2 -8 a 2 + z 2 + 2 z -2 + 3$ (db)
Kauffman polynomial	$a 6 z 8 + a 4 z 8 + a 7 z 7 + 2 a 5 z 7 + a 3 z 7 -6 a 6 z 6 -7 a 4 z 6 - a 2 z 6 -6 a 7 z 5 -14 a 5 z 5 -8 a 3 z 5 + 9 a 6 z 4 + 15 a 4 z 4 + 7 a 2 z 4 + z 4 + 10 a 7 z 3 + 28 a 5 z 3 + 20 a 3 z 3 + 2 a z 3 -4 a 6 z 2 -15 a 4 z 2 -15 a 2 z 2 -4 z 2 -6 a 7 z -19 a 5 z -21 a 3 z -8 a z + 3 a 6 + 10 a 4 + 11 a 2 + 5 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 L11n381. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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