L10a87

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_20,9,7,10 X₆₇₁₈ X_2,11,3,12 X_12,3,13,4 X_16,5,17,6 X_18,13,19,14 X_14,17,15,18 X_4,15,5,16 X_10,19,11,20
Gauss code	{1, -4, 5, -9, 6, -3}, {3, -1, 2, -10, 4, -5, 7, -8, 9, -6, 8, -7, 10, -2}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L10a87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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