L11n413

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_11,18,12,19 X_17,12,18,7 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, 9, -11, -8, 7, 10, -9, 11, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v w 2 u 2 - v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + v w 2 u + v u + v 2 w u -2 v w u + w u - w 2$ (db)
Jones polynomial	$- q 5 + 2 q 4 -3 q 3 + 4 q 2 -4 q + 4-2 q -1 + 2 q -2 + q -3 + q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 2 a 4 + 2 a 4 z -2 + 3 a 4 - z 4 a 2 -6 z 2 a 2 -5 a 2 z -2 -9 a 2 + z 4 + 3 z 2 + 4 z -2 + 6 + z 4 a -2 + 2 z 2 a -2 - a -2 z -2 + a -2 - z 2 a -4 - a -4$ (db)
Kauffman polynomial	$a 4 z 8 + a 2 z 8 + z 8 a -2 + z 8 + a 3 z 7 + 2 a z 7 + 3 z 7 a -1 + 2 z 7 a -3 -8 a 4 z 6 -10 a 2 z 6 -2 z 6 a -2 + 2 z 6 a -4 -6 z 6 -9 a 3 z 5 -16 a z 5 -14 z 5 a -1 -6 z 5 a -3 + z 5 a -5 + 21 a 4 z 4 + 31 a 2 z 4 - z 4 a -2 -6 z 4 a -4 + 15 z 4 + 22 a 3 z 3 + 41 a z 3 + 26 z 3 a -1 + 4 z 3 a -3 -3 z 3 a -5 -23 a 4 z 2 -39 a 2 z 2 + 3 z 2 a -2 + 3 z 2 a -4 -16 z 2 -19 a 3 z -35 a z -19 z a -1 -2 z a -3 + z a -5 + 11 a 4 + 22 a 2 - a -4 + 13 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

0 is the signature of L11n413. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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