L10n67

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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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 $u$ , $v$ , $w$ , ...)	$- v u 3 + 4 v u 2 - v w u 2 + 2 w u 2 - u 2 -2 v u + v w u -4 w u + u + w$ (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	$a 4 z 8 + a 2 z 8 + 2 a 5 z 7 + 6 a 3 z 7 + 4 a z 7 + a 6 z 6 + 3 a 4 z 6 + 6 a 2 z 6 + 4 z 6 -6 a 5 z 5 -18 a 3 z 5 -11 a z 5 + z 5 a -1 -4 a 6 z 4 -21 a 4 z 4 -30 a 2 z 4 -13 z 4 + 3 a 5 z 3 + 11 a 3 z 3 + 9 a z 3 + z 3 a -1 + 6 a 6 z 2 + 25 a 4 z 2 + 39 a 2 z 2 + 3 z 2 a -2 + 23 z 2 + a 5 z + a 3 z + a z + z a -1 -4 a 6 -14 a 4 -21 a 2 -4 a -2 -14- a 5 z -1 - a 3 z -1 - a z -1 - a -1 z -1 + a 6 z -2 + 3 a 4 z -2 + 4 a 2 z -2 + a -2 z -2 + 3 z -2$ (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 =

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

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