L6a4

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L6a3

L6a5

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

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

Visit L6a4's page at Knotilus.

Visit L6a4's page at the original Knot Atlas.

[edit L6a4 Quick Notes]

Also known as the "Borromean Link" or the "Borromean Rings". A Brunnian link - no two loops are linked directly together, but all three rings are collectively interlinked [8].

Visit Peter Cromwell's page on the Borromean Rings.

[edit L6a4 Further Notes and Views]

a kolam with 3 cycles [1]	A Borromean link by Dylan Thurston [2]	A "Borromean" bathroom tile (actually the Diane de Poitiers three crescents emblem) [3]	A Borromean rattle by Sassy [4]
A Borromean link at the Fields Institute [5]	Borromean paper clips [6]	The Colombo Mall in Lisboa [7]	Classic-type Borromean rings diagram with color-coded circles
The Borromean rings as a symbol of the Christian Trinity (based on a 13th-century French manuscript)	One version of the Germanic "Valknut"	A practical application of the Borromean rings (Ballard Locks, Seattle)	Coat of arms of Hallsberg, Sweden, with padlocks in Borromean configuration

[edit] Link Presentations

[edit Notes on L6a4's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,8,9,7 X_4,12,1,11 X_10,5,11,6 X₈₄₅₃ X_2,9,3,10
Gauss code	{1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u - v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$- q 3 + 3 q 2 -2 q + 4-2 q -1 + 3 q -2 - q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 4 - a 2 z 2 - z 2 a -2 + 2 z 2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 5 + 2 z 5 a -1 + 3 a 2 z 4 + 3 z 4 a -2 + 6 z 4 + a 3 z 3 - a z 3 - z 3 a -1 + z 3 a -3 -4 a 2 z 2 -4 z 2 a -2 -8 z 2 + 1-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (db)

[edit] Khovanov Homology

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 L6a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L6a4/KhovanovTable

Integral Khovanov Homology

(db, data source)

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