1.15 Borromean links
No two elements interlock, but all three do interlock. A three-component link with that property is called Borromean rings, after Borromeos, an Italian family from the Renaissance that used them as their family crest symbolizing the value of collaboration and unity. B. Lindström and H.O. Zetterström (1991) proved that Borromean circles are impossible: Borromean rings cannot be constructed from three flat circles, but can be constructed from three triangles. The Australian sculptor J. Robinson assembled three flat hollow triangles to form a structure (called Intuition), topologically equivalent to Borromean rings. Their cardboard model collapses under its own weight, to form a planar pattern. P. Cromwell recognized it in a picture-stone from Gotland (1995), and exactly these and other symmetrical combinations of three and four hollow triangles were considered by H.S.M. Coxeter (1994). In geometry, Borromean rings appear in a regular octahedron, in Venn diagrams, in DNA, and in other various areas.
In knot theory Borromean rings are the foremost examples having two remarkable properties: three mutually disjoint simple closed curves form a link, yet no two curves are linked, and if any one curve is cut, the other two are free to separate. In the case of 3-component links those two properties are inseparable: one follows from the other. In the case of n-component links ( n ³ 3), n-Borromean links could be defined as n-component non-trivial links such that any two components form a trivial link. Among them, those with at least one non-trivial sublink, for which we will keep the name Borromean links, will be distinguished from Brunnian links in which every sublink is trivial (Liang and Mislow, 1994c).
It seems surprising that besides the Borromean rings, represented by the link 623 in Rolfsen notation, no other link with the properties mentioned above can be found in link tables (Rolfsen, 1976; Adams, 1994). The reason for this is very simple: all existing knot tables contain only links with at most 9 crossings. In fact, an infinite number of n-Borromean or n-Brunnian links exist, and they can be derived as infinite series.