L10a66

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -3 v u 3 + u 3 -3 v 2 u 2 + 3 v u 2 -3 u 2 + v 2 u -3 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 + a 5 z 5 -5 a 3 z 5 + 2 a z 5 + 3 a 5 z 3 -9 a 3 z 3 + 8 a z 3 - z 3 a -1 + 2 a 5 z -8 a 3 z + 8 a z -3 z a -1 + a 5 z -1 -2 a 3 z -1 + 2 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 3 a 9 -3 z 4 a 8 + z 2 a 8 -5 z 5 a 7 + 4 z 3 a 7 - z a 7 -6 z 6 a 6 + 7 z 4 a 6 - z 2 a 6 -6 z 7 a 5 + 11 z 5 a 5 -5 z 3 a 5 + 2 z a 5 - a 5 z -1 -4 z 8 a 4 + 7 z 6 a 4 - z 9 a 3 -6 z 7 a 3 + 29 z 5 a 3 -31 z 3 a 3 + 12 z a 3 -2 a 3 z -1 -6 z 8 a 2 + 22 z 6 a 2 -22 z 4 a 2 + 7 z 2 a 2 - a 2 - z 9 a - z 7 a + 18 z 5 a -29 z 3 a + 14 z a -2 a z -1 -2 z 8 + 9 z 6 -12 z 4 + 5 z 2 - z 7 a -1 + 5 z 5 a -1 -8 z 3 a -1 + 5 z a -1 - a -1 z -1$ (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 L10a66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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