L10a96

(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_4,9,5,10 X_14,5,15,6 X_16,19,17,20 X_18,7,19,8 X_6,17,7,18 X_20,15,9,16 X_8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -8, 7, -10}, {4, -1, 2, -3, 10, -5, 9, -6, 8, -7, 6, -9}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L10a96. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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