L10a86

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_18,10,19,9 X_14,19,15,20 X_10,16,11,15 X_20,11,7,12 X₄₇₅₈ X_2,14,3,13 X_12,4,13,3 X_6,18,1,17
Gauss code	{1, -8, 9, -7, 2, -10}, {7, -1, 3, -5, 6, -9, 8, -4, 5, -2, 10, -3, 4, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 3 v 2 u 3 -7 v u 3 + 3 u 3 -4 v 2 u 2 + 11 v u 2 -4 u 2 + 3 v 2 u -7 v u + 3 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+13 q^{7/2}-17 q^{5/2}+18 q^{3/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -4 z 5 a -1 + 2 z 5 a -3 + 2 a z 3 -7 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + 2 a z -5 z a -1 + 5 z a -3 - z a -5 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-3 z 9 a -1 -3 z 9 a -3 -16 z 8 a -2 -8 z 8 a -4 -8 z 8 -8 a z 7 -12 z 7 a -1 -12 z 7 a -3 -8 z 7 a -5 -4 a 2 z 6 + 30 z 6 a -2 + 11 z 6 a -4 -4 z 6 a -6 + 11 z 6 - a 3 z 5 + 14 a z 5 + 36 z 5 a -1 + 36 z 5 a -3 + 14 z 5 a -5 - z 5 a -7 + 5 a 2 z 4 -18 z 4 a -2 -4 z 4 a -4 + 5 z 4 a -6 -4 z 4 + a 3 z 3 -9 a z 3 -31 z 3 a -1 -31 z 3 a -3 -9 z 3 a -5 + z 3 a -7 - a 2 z 2 + 4 z 2 a -2 + z 2 a -4 - z 2 a -6 + z 2 + 4 a z + 12 z a -1 + 12 z a -3 + 4 z a -5 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 =

1 is the signature of L10a86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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