L10n58

(Knotscape image)

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Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_17,11,18,20 X_13,19,14,18 X_19,15,20,14 X_2,11,3,12 X_15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 - v 3 u 2 + 4 v 2 u 2 -4 v u 2 + u 2 + v 3 u -4 v 2 u + 4 v u - u + v 2 - v$ (db)
Jones polynomial	$- q 19 / 2 + 3 q 17 / 2 -5 q 15 / 2 + 8 q 13 / 2 -8 q 11 / 2 + 8 q 9 / 2 -8 q 7 / 2 + 4 q 5 / 2 -3 q 3 / 2$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2 z 5 a -5 + 3 z 3 a -3 -8 z 3 a -5 + 3 z 3 a -7 + 7 z a -3 -13 z a -5 + 7 z a -7 - z a -9 + 4 a -3 z -1 -8 a -5 z -1 + 5 a -7 z -1 - a -9 z -1$ (db)
Kauffman polynomial	$- z 8 a -6 - z 8 a -8 -4 z 7 a -5 -7 z 7 a -7 -3 z 7 a -9 -3 z 6 a -4 -8 z 6 a -6 -8 z 6 a -8 -3 z 6 a -10 + 10 z 5 a -5 + 14 z 5 a -7 + 3 z 5 a -9 - z 5 a -11 + 6 z 4 a -4 + 25 z 4 a -6 + 26 z 4 a -8 + 7 z 4 a -10 -6 z 3 a -3 -20 z 3 a -5 -12 z 3 a -7 + 4 z 3 a -9 + 2 z 3 a -11 -12 z 2 a -4 -31 z 2 a -6 -24 z 2 a -8 -5 z 2 a -10 + 10 z a -3 + 19 z a -5 + 9 z a -7 - z a -9 - z a -11 + 8 a -4 + 14 a -6 + 9 a -8 + 2 a -10 -4 a -3 z -1 -8 a -5 z -1 -5 a -7 z -1 - a -9 z -1$ (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 L10n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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