L10a68

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -3 v u 3 + u 3 -3 v 2 u 2 + 5 v u 2 -3 u 2 + v 2 u -3 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+4 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 9 a z 3 -8 z 3 a -1 + z 3 a -3 -2 a 3 z + 7 a z -9 z a -1 + 3 z a -3 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -2 z 8 a -2 -5 z 8 -4 a 3 z 7 -4 a z 7 -2 z 7 a -1 -2 z 7 a -3 -4 a 4 z 6 + 3 a 2 z 6 + 4 z 6 a -2 - z 6 a -4 + 12 z 6 -3 a 5 z 5 + 4 a 3 z 5 + 14 a z 5 + 14 z 5 a -1 + 7 z 5 a -3 - a 6 z 4 + 4 a 4 z 4 - a 2 z 4 + 3 z 4 a -2 + 4 z 4 a -4 -7 z 4 + 4 a 5 z 3 - a 3 z 3 -17 a z 3 -19 z 3 a -1 -7 z 3 a -3 + a 6 z 2 - a 2 z 2 -7 z 2 a -2 -4 z 2 a -4 -3 z 2 - a 5 z + 9 a z + 12 z a -1 + 4 z a -3 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L10a68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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