L10a50

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,5,11,6 X_14,3,15,4 X_20,11,5,12 X_18,13,19,14 X_12,19,13,20 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 -6 v u 2 + 7 u 2 + 7 v u -6 u -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 11 - a 11 z -1 - z 3 a 9 + 2 z a 9 + 2 a 9 z -1 -3 z 3 a 7 -2 z a 7 -3 z 3 a 5 -3 z a 5 - a 5 z -1 - z 3 a 3$ (db)
Kauffman polynomial	$- z 7 a 13 + 4 z 5 a 13 -4 z 3 a 13 + z a 13 -3 z 8 a 12 + 14 z 6 a 12 -19 z 4 a 12 + 6 z 2 a 12 + 2 a 12 -2 z 9 a 11 + 5 z 7 a 11 + 3 z 5 a 11 -9 z 3 a 11 + 2 z a 11 - a 11 z -1 -8 z 8 a 10 + 30 z 6 a 10 -29 z 4 a 10 + 2 z 2 a 10 + 5 a 10 -2 z 9 a 9 - z 7 a 9 + 17 z 5 a 9 -15 z 3 a 9 + 4 z a 9 -2 a 9 z -1 -5 z 8 a 8 + 9 z 6 a 8 + z 4 a 8 -4 z 2 a 8 + 3 a 8 -7 z 7 a 7 + 12 z 5 a 7 -3 z 3 a 7 -7 z 6 a 6 + 8 z 4 a 6 - a 6 -6 z 5 a 5 + 6 z 3 a 5 -3 z a 5 + a 5 z -1 -3 z 4 a 4 - z 3 a 3$ (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 L10a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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