L10n65

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_11,16,12,17 X_13,19,14,18 X_17,20,18,9 X_19,13,20,12 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
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 $u$ , $v$ , $w$ , ...)	$\frac{u v w^3-3 u v w^2+3 u v w-u v-2 u w+u-v w^3+2 v w^2+w^3-3 w^2+3 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$3 q -5 -5 q -4 + q 3 + 7 q -3 -2 q 2 -7 q -2 + 5 q + 8 q -1 -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 6 z -2 + a 6 - z 4 a 4 -3 z 2 a 4 -3 a 4 z -2 -5 a 4 + z 6 a 2 + 4 z 4 a 2 + 7 z 2 a 2 + 4 a 2 z -2 + 8 a 2 -2 z 4 -6 z 2 -3 z -2 -6 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 2 z 8 + z 8 + 5 a 3 z 7 + 7 a z 7 + 2 z 7 a -1 + 7 a 4 z 6 + 10 a 2 z 6 + z 6 a -2 + 4 z 6 + 3 a 5 z 5 -9 a 3 z 5 -18 a z 5 -6 z 5 a -1 -18 a 4 z 4 -37 a 2 z 4 -4 z 4 a -2 -23 z 4 + 4 a 3 z 3 + 8 a z 3 + 4 z 3 a -1 + 6 a 6 z 2 + 24 a 4 z 2 + 40 a 2 z 2 + 6 z 2 a -2 + 28 z 2 + a 5 z + a 3 z + a z + z a -1 -4 a 6 -14 a 4 -21 a 2 -4 a -2 -14- a 5 z -1 - a 3 z -1 - a z -1 - a -1 z -1 + a 6 z -2 + 3 a 4 z -2 + 4 a 2 z -2 + a -2 z -2 + 3 z -2$ (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 =

-2 is the signature of L10n65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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