L10a89

(Knotscape image)

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_14,6,15,5 X_2,9,3,10 X_4,14,5,13 X_18,16,19,15 X_16,7,17,8 X_6,17,7,18 X_8,20,1,19
Gauss code	{1, -5, 2, -6, 4, -9, 8, -10}, {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + 2 v 3 u 2 -3 v 2 u 2 + 2 v u 2 - u 2 - v 3 u + 2 v 2 u -3 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+7 q^{5/2}-7 q^{3/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 8 z 3 a -3 -3 z 3 a -5 + 3 a z -7 z a -1 + 5 z a -3 - z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -5 z 8 a -2 -3 z 8 a -4 -2 z 8 - a z 7 -3 z 7 a -3 -4 z 7 a -5 + 18 z 6 a -2 + 5 z 6 a -4 -4 z 6 a -6 + 9 z 6 + 5 a z 5 + 14 z 5 a -1 + 18 z 5 a -3 + 5 z 5 a -5 -4 z 5 a -7 -15 z 4 a -2 + z 4 a -4 + 2 z 4 a -6 -3 z 4 a -8 -11 z 4 -8 a z 3 -23 z 3 a -1 -17 z 3 a -3 + 2 z 3 a -5 + 3 z 3 a -7 - z 3 a -9 + z 2 a -2 - z 2 a -4 + 3 z 2 a -6 + 2 z 2 a -8 + 3 z 2 + 5 a z + 10 z a -1 + 4 z a -3 -2 z a -5 - z a -7 + 1- a z -1 - a -1 z -1$ (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 =

3 is the signature of L10a89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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