L9a50

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,16,11,15 X_14,8,15,7 X_10,12,5,11 X_8,17,9,18 X_16,9,17,10 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 -2 v 2 u + 2 v u + v 2 w u -2 v w u + 2 w u - u + v 2 -2 v + v w - w + 1$ (db)
Jones polynomial	$- q 6 + 3 q 5 -5 q 4 + 7 q 3 -7 q 2 + 8 q -5 + 5 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 4 z 4 a -2 - z 4 a -4 -2 z 4 + a 2 z 2 + 6 z 2 a -2 -2 z 2 a -4 -6 z 2 + 2 a 2 + 4 a -2 - a -4 -5 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 6 z 7 a -1 + 4 z 7 a -3 + a 2 z 6 + 8 z 6 a -2 + 6 z 6 a -4 + 3 z 6 -6 a z 5 -14 z 5 a -1 -3 z 5 a -3 + 5 z 5 a -5 -4 a 2 z 4 -27 z 4 a -2 -9 z 4 a -4 + 3 z 4 a -6 -19 z 4 + 3 a z 3 + 3 z 3 a -1 -5 z 3 a -3 -4 z 3 a -5 + z 3 a -7 + 6 a 2 z 2 + 23 z 2 a -2 + 6 z 2 a -4 - z 2 a -6 + 22 z 2 + 3 a z + 5 z a -1 + 3 z a -3 + z a -5 -4 a 2 -8 a -2 -2 a -4 -9-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L9a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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