L10n89

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_20,15,17,16 X_18,11,19,12 X_12,17,13,18 X_16,19,5,20 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(3)-1) \left(t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$2 q -10 -4 q -9 + 7 q -8 -8 q -7 + 9 q -6 -7 q -5 + 7 q -4 -3 q -3 + q -2$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 10 z -2 + 2 a 10 -5 z 2 a 8 -2 a 8 z -2 -8 a 8 + 3 z 4 a 6 + 8 z 2 a 6 + a 6 z -2 + 6 a 6 + z 4 a 4 + z 2 a 4$ (db)
Kauffman polynomial	$3 a 12 z 4 -5 a 12 z 2 + 2 a 12 + a 11 z 7 + a 11 z 5 -2 a 11 z 3 + a 10 z 8 + 2 a 10 z 6 -3 a 10 z 4 + a 10 z -2 -2 a 10 + 5 a 9 z 7 -3 a 9 z 5 -8 a 9 z 3 + 8 a 9 z -2 a 9 z -1 + a 8 z 8 + 8 a 8 z 6 -18 a 8 z 4 + 16 a 8 z 2 + 2 a 8 z -2 -9 a 8 + 4 a 7 z 7 - a 7 z 5 -8 a 7 z 3 + 8 a 7 z -2 a 7 z -1 + 6 a 6 z 6 -11 a 6 z 4 + 10 a 6 z 2 + a 6 z -2 -6 a 6 + 3 a 5 z 5 -2 a 5 z 3 + a 4 z 4 - a 4 z 2$ (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 =

-4 is the signature of L10n89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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