L10a59

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,13,17,14 X_20,18,7,17 X_18,12,19,11 X_12,20,13,19 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 6, -7, 4, -3, 8, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 - v 2 u 3 + 5 v u 3 -3 u 3 + 3 v 2 u 2 -7 v u 2 + 3 u 2 -3 v 2 u + 5 v u - u + v 2 - v$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+6 q^{7/2}-9 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 + z 5 a -1 - a 3 z 3 + 2 a z 3 -2 z 3 a -3 - a 3 z + 3 a z -3 z a -1 - z a -3 + z a -5 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -3 z 8 a -2 -6 z 8 -3 a 3 z 7 -6 a z 7 -8 z 7 a -1 -5 z 7 a -3 - a 4 z 6 + 6 a 2 z 6 -4 z 6 a -2 -5 z 6 a -4 + 8 z 6 + 10 a 3 z 5 + 23 a z 5 + 20 z 5 a -1 + 4 z 5 a -3 -3 z 5 a -5 + 3 a 4 z 4 + a 2 z 4 + 15 z 4 a -2 + 6 z 4 a -4 - z 4 a -6 + 6 z 4 -10 a 3 z 3 -23 a z 3 -17 z 3 a -1 - z 3 a -3 + 3 z 3 a -5 -2 a 4 z 2 -4 a 2 z 2 -12 z 2 a -2 -4 z 2 a -4 + z 2 a -6 -9 z 2 + 3 a 3 z + 11 a z + 10 z a -1 + z a -3 - z a -5 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L10a59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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