Are Borromean Links So Rare
Václav Slavík · Forma · 2000
After describing classical Borromean links and their properties, Borromean property and Brunnean property are extended to n-component links (n > 3). Different known infinite classes of three-component Borromean links are described, as well as two new infinite classes of “prismatic” Borromean n-component links. “No two elements interlock, but all three do interlock”. A three-component link with that property is called “Borromean” after the Borromeos, an Italian family from the Renaissance that used them as their family crest symbolizing the value of collaboration and unity. B. Lindstrom and H. O. Zetterstrom (LINDSTROM and ZETTERSTROM, 1991), proved that “Borromean circles are impossible”: three flat circles cannot construct them, but by triangles they can. 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. Cromwel recognized Borromean triangles in a picture-stone from Gotland (CROMWEL, 1995). This and other symmetrical combinations of three and four hollow triangles were considered by H. S. M. Coxeter (COXETER, 1994). In geometry, Borromean rings appear as the regular octahedron {3,4} (JABLAN, 1998), in Venn diagrams (RUSKEY, 1999), in DNA (SEEMAN, 1999), and in other various areas (CROMWEL et al., 1998) (Fig. 1). In the knot theory Borromean rings are the foremost examples having with 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), n-Borromean links could be defined as n-component nontrivial links such that any two components form a trivial link. Among them, those with at least one nontrivial sublink, for which we will keep the name “Borromean links”, will be distinguished from the Brunnian links in which every sublink is trivial (LIANG and MISLOW, 1994). It seams surprising that besides the Borromean rings, represented by the link 62 3 in Rolfsen’s notation, no other link with the properties mentioned above can be found in link tables (ROLFSEN, 1990; ADAMS, 1994). The reason for this is very simple: all existing knot