L10n67

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,17 X_13,19,14,18 X_17,20,18,9 X_19,13,20,12 X_8,16,5,15 X_14,8,15,7 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(2) t(3)^3-2 t(1) t(3)^2+t(1) t(2) t(3)^2-4 t(2) t(3)^2+t(3)^2+4 t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$2 q 2 -3 q + 6-6 q -1 + 6 q -2 -5 q -3 + 5 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 z -2 + a 6 -3 z 2 a 4 -3 a 4 z -2 -5 a 4 + 2 z 4 a 2 + 6 z 2 a 2 + 4 a 2 z -2 + 8 a 2 -4 z 2 -3 z -2 -6 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 6 z 6 -4 a 6 z 4 + 6 a 6 z 2 + a 6 z -2 -4 a 6 + 2 a 5 z 7 -6 a 5 z 5 + 3 a 5 z 3 + a 5 z - a 5 z -1 + a 4 z 8 + 3 a 4 z 6 -21 a 4 z 4 + 25 a 4 z 2 + 3 a 4 z -2 -14 a 4 + 6 a 3 z 7 -18 a 3 z 5 + 11 a 3 z 3 + a 3 z - a 3 z -1 + a 2 z 8 + 6 a 2 z 6 -30 a 2 z 4 + 39 a 2 z 2 + 3 z 2 a -2 + 4 a 2 z -2 + a -2 z -2 -21 a 2 -4 a -2 + 4 a z 7 -11 a z 5 + z 5 a -1 + 9 a z 3 + z 3 a -1 + a z + z a -1 - a z -1 - a -1 z -1 + 4 z 6 -13 z 4 + 23 z 2 + 3 z -2 -14$ (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 =

0 is the signature of L10n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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