L10a58

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 - v 2 u 3 + 5 v u 3 -2 u 3 + 4 v 2 u 2 -7 v u 2 + 4 u 2 -2 v 2 u + 5 v u - u - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 -2 z 3 a 7 -3 z a 7 -2 a 7 z -1 + z 5 a 5 + z 3 a 5 + z a 5 + 2 a 5 z -1 + z 5 a 3 + z 3 a 3 - z a 3 - a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -2 z a 11 -2 z 6 a 10 + 4 z 4 a 10 - z 2 a 10 -3 z 7 a 9 + 6 z 5 a 9 -5 z 3 a 9 + 4 z a 9 - a 9 z -1 -3 z 8 a 8 + 6 z 6 a 8 -8 z 4 a 8 + 5 z 2 a 8 - z 9 a 7 -5 z 7 a 7 + 17 z 5 a 7 -23 z 3 a 7 + 11 z a 7 -2 a 7 z -1 -6 z 8 a 6 + 13 z 6 a 6 -15 z 4 a 6 + 7 z 2 a 6 - a 6 - z 9 a 5 -6 z 7 a 5 + 18 z 5 a 5 -20 z 3 a 5 + 9 z a 5 -2 a 5 z -1 -3 z 8 a 4 + 2 z 6 a 4 + 3 z 4 a 4 - z 2 a 4 -4 z 7 a 3 + 7 z 5 a 3 -3 z 3 a 3 + 3 z a 3 - a 3 z -1 -3 z 6 a 2 + 6 z 4 a 2 -2 z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L10a58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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