L10a67

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 4 v 2 u 3 -2 v u 3 -2 v 2 u 2 + 5 v u 2 -2 u 2 -2 v u + 4 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 9 + 3 z a 9 + a 9 z -1 -2 z 5 a 7 -8 z 3 a 7 -8 z a 7 -2 a 7 z -1 + z 7 a 5 + 5 z 5 a 5 + 8 z 3 a 5 + 6 z a 5 + 2 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -4 z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 + z a 13 -2 z 4 a 12 + z 2 a 12 -3 z 5 a 11 + 2 z 3 a 11 - z a 11 -4 z 6 a 10 + 6 z 4 a 10 -4 z 2 a 10 -4 z 7 a 9 + 8 z 5 a 9 -6 z 3 a 9 + 4 z a 9 - a 9 z -1 -3 z 8 a 8 + 6 z 6 a 8 - z 4 a 8 - z 9 a 7 -3 z 7 a 7 + 20 z 5 a 7 -24 z 3 a 7 + 12 z a 7 -2 a 7 z -1 -5 z 8 a 6 + 19 z 6 a 6 -20 z 4 a 6 + 8 z 2 a 6 - a 6 - z 9 a 5 + 14 z 5 a 5 -23 z 3 a 5 + 11 z a 5 -2 a 5 z -1 -2 z 8 a 4 + 9 z 6 a 4 -11 z 4 a 4 + 3 z 2 a 4 - z 7 a 3 + 5 z 5 a 3 -8 z 3 a 3 + 5 z a 3 - a 3 z -1$ (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 =

-5 is the signature of L10a67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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