L11n390

(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 X₃₈₄₉ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_11,18,12,5 X_15,2,16,3
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 $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 3 - v w u 3 + w u 3 - u 3 - v u 2 + v w u 2 -2 w u 2 + u 2 + w$ (db)
Jones polynomial	$q 2 + 1 + 2 q -1 -2 q -2 + 4 q -3 -3 q -4 + 3 q -5 -3 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 6 - a 6 z -2 - a 6 + 4 z 2 a 4 + 4 a 4 z -2 + 8 a 4 - z 6 a 2 -7 z 4 a 2 -14 z 2 a 2 -5 a 2 z -2 -13 a 2 + z 4 + 5 z 2 + 2 z -2 + 6$ (db)
Kauffman polynomial	$z 4 a 8 - z 2 a 8 + 3 z 5 a 7 -6 z 3 a 7 + a 7 z -1 + 2 z 6 a 6 -3 z 4 a 6 - a 6 z -2 + a 6 + 3 z 5 a 5 -9 z a 5 + 5 a 5 z -1 + 6 z 4 a 4 -13 z 2 a 4 -4 a 4 z -2 + 12 a 4 + z 7 a 3 -9 z 5 a 3 + 28 z 3 a 3 -27 z a 3 + 9 a 3 z -1 + z 8 a 2 -10 z 6 a 2 + 31 z 4 a 2 -37 z 2 a 2 -5 a 2 z -2 + 21 a 2 + z 7 a -9 z 5 a + 22 z 3 a -18 z a + 5 a z -1 + z 8 -8 z 6 + 21 z 4 -23 z 2 -2 z -2 + 11$ (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 L11n390. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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