L10n91

(Knotscape image)

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Visit L10n91's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,20,14,17 X_7,18,8,19 X_17,10,18,11 X_9,15,10,14 X_15,9,16,8 X_19,16,20,5 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(3)-1) (t(3)+1) (-2 t(1) t(2)+t(1) t(3) t(2)-2 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$1- q -1 + q -2 - q -3 + q -5 + q -6 + 2 q -7 - q -8 + q -9$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 10 z -2 -2 a 8 z -2 -4 a 8 + z 4 a 6 + 6 z 2 a 6 + a 6 z -2 + 6 a 6 - z 6 a 4 -5 z 4 a 4 -6 z 2 a 4 -4 a 4 + z 4 a 2 + 4 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$a 10 z 6 -5 a 10 z 4 + 6 a 10 z 2 + a 10 z -2 -4 a 10 + a 9 z 7 -4 a 9 z 5 + 4 a 9 z -2 a 9 z -1 + a 8 z 8 -5 a 8 z 6 + 3 a 8 z 4 + 6 a 8 z 2 + 2 a 8 z -2 -5 a 8 + 2 a 7 z 7 -12 a 7 z 5 + 16 a 7 z 3 -4 a 7 z -2 a 7 z -1 + a 6 z 8 -5 a 6 z 6 + 2 a 6 z 4 + 6 a 6 z 2 + a 6 z -2 -2 a 6 + 2 a 5 z 7 -13 a 5 z 5 + 22 a 5 z 3 -12 a 5 z + 2 a 4 z 6 -11 a 4 z 4 + 12 a 4 z 2 -2 a 4 + a 3 z 7 -5 a 3 z 5 + 6 a 3 z 3 -4 a 3 z + a 2 z 6 -5 a 2 z 4 + 6 a 2 z 2 -2 a 2$ (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 =

-3 is the signature of L10n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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