Borromean ring

Borromean rings
L6a4
Braid length	6
Braid no.	3
Crossing no.	6
Hyperbolic volume	7.327724753
Stick no.	9
Unknotting no.	2
Conway notation	[.1]
A-B notation	6³ ₂
Thistlethwaite	L6a4
Last /Next	L6a3 / L6a5
Other
alternating, hyperbolic

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings). In other words, no two of the three rings are linked with each other as a Hopf link, but nonetheless all three are linked.

Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circular rings, they cannot be. Freedman and Skora (1987) prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see (Lindström & Zetterström 1991).

It is, however, true that one can use ellipses (right picture). These may be taken to be of arbitrarily small eccentricity; i.e. no matter how close to being circular their shape may be, as long as they are not perfectly circular, they can form Borromean links if suitably positioned; as an example, thin circles made from bendable elastic wire may be used as Borromean rings.

Apart from indicating which strand crosses over the other, link diagrams use the same notation to show two strands crossing, as graph diagrams use to show four edges meeting at a common vertex. Accordingly, the graph of the regular octahedron may be converted into a link diagram by prescribing that, as a strand follows successive edges, it alternates between passing over a vertex and passing under the next. The result has three separate loops, linked together as Borromean rings.

In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. There are a number of ways of seeing this.

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Wikipedia