L10a42

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 9 v u 2 -8 u 2 -8 v u + 9 u + 2 v -2$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{10}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 -2 z 3 a 7 - z a 7 + z 5 a 5 + a 5 z -1 + z 5 a 3 -2 z a 3 - a 3 z -1 - z 3 a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 6 z 4 a 10 -3 z 2 a 10 -4 z 7 a 9 + 5 z 5 a 9 -4 z 8 a 8 + 4 z 6 a 8 - z 4 a 8 + z 2 a 8 -2 z 9 a 7 -4 z 7 a 7 + 11 z 5 a 7 -8 z 3 a 7 + z a 7 -10 z 8 a 6 + 21 z 6 a 6 -19 z 4 a 6 + 6 z 2 a 6 -2 z 9 a 5 -7 z 7 a 5 + 20 z 5 a 5 -14 z 3 a 5 + 2 z a 5 - a 5 z -1 -6 z 8 a 4 + 10 z 6 a 4 -6 z 4 a 4 + 2 z 2 a 4 + a 4 -7 z 7 a 3 + 14 z 5 a 3 -7 z 3 a 3 + 2 z a 3 - a 3 z -1 -4 z 6 a 2 + 6 z 4 a 2 - z 5 a + z 3 a$ (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 L10a42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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