L10a98

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + v 2 u 3 + v 3 u 2 -3 v 2 u 2 + v u 2 + v 2 u -3 v u + u + v -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{3}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z 5 a 11 + 4 z 3 a 11 + 3 z a 11 - z 7 a 9 -5 z 5 a 9 -6 z 3 a 9 + a 9 z -1 - z 7 a 7 -6 z 5 a 7 -11 z 3 a 7 -7 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 3 a 17 + z a 17 -2 z 4 a 16 + 2 z 2 a 16 -2 z 5 a 15 + z 3 a 15 -2 z 6 a 14 + 2 z 4 a 14 - z 2 a 14 -2 z 7 a 13 + 4 z 5 a 13 -3 z 3 a 13 + z a 13 -2 z 8 a 12 + 7 z 6 a 12 -9 z 4 a 12 + 5 z 2 a 12 - z 9 a 11 + 3 z 7 a 11 -3 z 5 a 11 + 5 z 3 a 11 -3 z a 11 -3 z 8 a 10 + 13 z 6 a 10 -15 z 4 a 10 + 5 z 2 a 10 - z 9 a 9 + 4 z 7 a 9 -3 z 5 a 9 - z 3 a 9 + 2 z a 9 - a 9 z -1 - z 8 a 8 + 4 z 6 a 8 -2 z 4 a 8 -3 z 2 a 8 + a 8 - z 7 a 7 + 6 z 5 a 7 -11 z 3 a 7 + 7 z a 7 - a 7 z -1$ (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 =

-7 is the signature of L10a98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$
$r = -10$	${\mathbb Z}$
$r = -9$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$