L10n39

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,10,15,9 X_16,12,17,11 X_10,16,11,15 X_17,5,18,20 X_7,19,8,18 X_19,9,20,8 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 3, -5, 4, -2, 10, -3, 5, -4, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + v u 4 - u 4 - v u 3 + u 3 + v u 2 - u 2 - v u + u + v -1$ (db)
Jones polynomial	$-q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 + z 5 a -3 -6 z 5 a -5 + z 5 a -7 + 5 z 3 a -3 -13 z 3 a -5 + 5 z 3 a -7 + 8 z a -3 -15 z a -5 + 8 z a -7 - z a -9 + 4 a -3 z -1 -8 a -5 z -1 + 5 a -7 z -1 - a -9 z -1$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - z 7 a -3 -5 z 7 a -5 -4 z 7 a -7 + 3 z 6 a -4 -2 z 6 a -6 -5 z 6 a -8 + 6 z 5 a -3 + 24 z 5 a -5 + 16 z 5 a -7 -2 z 5 a -9 + 3 z 4 a -4 + 23 z 4 a -6 + 20 z 4 a -8 -13 z 3 a -3 -35 z 3 a -5 -18 z 3 a -7 + 4 z 3 a -9 -13 z 2 a -4 -33 z 2 a -6 -23 z 2 a -8 -3 z 2 a -10 + 12 z a -3 + 23 z a -5 + 11 z a -7 - z a -9 - z a -11 + 8 a -4 + 14 a -6 + 9 a -8 + 2 a -10 -4 a -3 z -1 -8 a -5 z -1 -5 a -7 z -1 - a -9 z -1$ (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 =

5 is the signature of L10n39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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