L9a42

(Knotscape image)

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_18,5,9,6 X_6,9,7,10 X_16,12,17,11 X_14,8,15,7 X_4,14,5,13 X_8,16,1,15 X_2,17,3,18
Gauss code	{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + v 3 u 2 -5 v 2 u 2 + 4 v u 2 + 4 v 2 u -5 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+6 q^{7/2}-9 q^{5/2}+9 q^{3/2}-10 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -5 z 5 a -1 + z 5 a -3 + 3 a z 3 -9 z 3 a -1 + 3 z 3 a -3 + 3 a z -6 z a -1 + 3 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 8 a -2 -2 z 8 -4 a z 7 -9 z 7 a -1 -5 z 7 a -3 -3 a 2 z 6 -5 z 6 a -2 -5 z 6 a -4 -3 z 6 - a 3 z 5 + 7 a z 5 + 17 z 5 a -1 + 6 z 5 a -3 -3 z 5 a -5 + 6 a 2 z 4 + 12 z 4 a -2 + 6 z 4 a -4 - z 4 a -6 + 11 z 4 + 2 a 3 z 3 -3 a z 3 -12 z 3 a -1 -4 z 3 a -3 + 3 z 3 a -5 -3 a 2 z 2 -6 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -5 z 2 - a 3 z + 2 a z + 6 z a -1 + 2 z a -3 - z a -5 + 1- a z -1 - a -1 z -1$ (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 L9a42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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