L10n107

(Knotscape image)

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

L10n107 is the "Borromean chain mail" link - it contains two L6a4 configurations without any L2a1 configuration (i.e. no two loops are linked).

An indefinitely extended "Borromean chainmail" pattern made up of overlapping L10n107 knots; no two circles are directly linked.

"Borromean chain-mail" represented with circles

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,11,10,14 X_13,15,14,20 X_19,5,20,10 X_11,18,12,19 X_4,17,1,18
Gauss code	{1, -4, 3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, -3, -6, 8}, {4, -5, 10, 9, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+q^{5/2}-2 q^{3/2}-2 \sqrt{q}-\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a z 5 - z 5 a -1 - a 3 z 3 + 5 a z 3 -5 z 3 a -1 + z 3 a -3 -2 a 3 z + 6 a z -6 z a -1 + 2 z a -3 + a 3 z -3 -3 a z -3 + 3 a -1 z -3 - a -3 z -3$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 a -2 -2 z 8 -2 a 3 z 7 -4 a z 7 -4 z 7 a -1 -2 z 7 a -3 - a 4 z 6 + 4 a 2 z 6 + 4 z 6 a -2 - z 6 a -4 + 10 z 6 + 10 a 3 z 5 + 26 a z 5 + 26 z 5 a -1 + 10 z 5 a -3 + 4 a 4 z 4 + 2 a 2 z 4 + 2 z 4 a -2 + 4 z 4 a -4 -4 z 4 -12 a 3 z 3 -44 a z 3 -44 z 3 a -1 -12 z 3 a -3 -2 a 4 z 2 -8 a 2 z 2 -8 z 2 a -2 -2 z 2 a -4 -12 z 2 + 8 a 3 z + 24 a z + 24 z a -1 + 8 z a -3 + 1-3 a z -1 -3 a -1 z -1 + 3 a 2 z -2 + 3 a -2 z -2 + 6 z -2 - a 3 z -3 -3 a z -3 -3 a -1 z -3 - a -3 z -3$ (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 L10n107. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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