L11n414

(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_18,12,19,11 X_12,18,7,17 X_15,20,16,21 X_19,22,20,13 X_21,16,22,17
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 $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v 2 u 2 - w 2 u 2 + v 2 u + w 2 u - v w 2 - v 2 w + v w$ (db)
Jones polynomial	$q 3 - q 2 + 2 q -1 + 2 q -1 + q -3 + q -4 - q -5 + q -6 - q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + z 4 a 4 + 5 z 2 a 4 + 4 a 4 z -2 + 8 a 4 - z 4 a 2 -5 z 2 a 2 -5 a 2 z -2 -8 a 2 - z 4 -3 z 2 + 2 z -2 + z 2 a -2 + 2 a -2$ (db)
Kauffman polynomial	$a 6 z 8 + a 4 z 8 + a 2 z 8 + z 8 + a 7 z 7 + 2 a 5 z 7 + 2 a 3 z 7 + 2 a z 7 + z 7 a -1 -6 a 6 z 6 -8 a 4 z 6 -8 a 2 z 6 + z 6 a -2 -5 z 6 -6 a 7 z 5 -14 a 5 z 5 -16 a 3 z 5 -12 a z 5 -4 z 5 a -1 + 10 a 6 z 4 + 20 a 4 z 4 + 22 a 2 z 4 -5 z 4 a -2 + 7 z 4 + 10 a 7 z 3 + 28 a 5 z 3 + 37 a 3 z 3 + 21 a z 3 + 2 z 3 a -1 -7 a 6 z 2 -25 a 4 z 2 -31 a 2 z 2 + 6 z 2 a -2 -7 z 2 -5 a 7 z -21 a 5 z -33 a 3 z -16 a z + z a -1 + 4 a 6 + 17 a 4 + 20 a 2 -2 a -2 + 6 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 =

-2 is the signature of L11n414. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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