L10a99

(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_2,9,3,10 X_18,14,19,13 X_14,7,15,8 X_16,5,17,6 X_6,15,7,16 X_4,17,5,18 X_8,20,1,19
Gauss code	{1, -4, 2, -9, 7, -8, 6, -10}, {4, -1, 3, -2, 5, -6, 8, -7, 9, -5, 10, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + 2 v u 3 - u 3 - v 3 u 2 + 3 v 2 u 2 -3 v u 2 + 2 u 2 + 2 v 3 u -3 v 2 u + 3 v u - u - v 3 + 2 v 2 - v$ (db)
Jones polynomial	$-q^{9/2}+3 q^{7/2}-5 q^{5/2}+7 q^{3/2}-8 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -4 z a 3 - a 3 z -1 + z 5 a + 2 z 3 a + z a + z 5 a -1 + 2 z 3 a -1 + z a -1 - z 3 a -3 - z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -5 a 2 z 8 -3 z 8 - a 5 z 7 -4 a z 7 -5 z 7 a -1 + 9 a 4 z 6 + 17 a 2 z 6 -6 z 6 a -2 + 2 z 6 + 5 a 5 z 5 + 13 a 3 z 5 + 20 a z 5 + 7 z 5 a -1 -5 z 5 a -3 -11 a 4 z 4 -14 a 2 z 4 + 8 z 4 a -2 -3 z 4 a -4 + 8 z 4 -8 a 5 z 3 -21 a 3 z 3 -19 a z 3 - z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + 2 a 4 z 2 + 2 a 2 z 2 -3 z 2 a -2 + z 2 a -4 -4 z 2 + 5 a 5 z + 9 a 3 z + 6 a z + z a -1 - z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L10a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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