The first such series
of 3-component links, beginning with the Borromean rings, was discovered
by P.G. Tait (1876-77). Their geometrical source is easy to recognize:
the limiting case for
If it is not necessary that every two components do intersect, an infinite number of "fractal" Borromean links can be derived from each n-Borromean link in a very simple way. Indeed, it is enough to surround an even number of the appropriately chosen crossing points of any two components by circles. Therefore, our consideration will be restricted to n-Borromean links without non-intersecting components.
The next infinite series of 3-Borromean links, beginning again with the Borromean rings, follow from the family of 2-component links (4n-2)12 (212, 612, 1012 ...) by introducing the third component: a circle intersecting opposite bigons (a). In a similar way, from the family of 2-component links (2n)12 (212, 412, 612 ...) we derive another infinite series of 3-component Borromean links without bigons (b). From such links with a self-intersecting component, new infinite series are obtained. In a self-crossing point of the oriented component an even chain of bigons is introduced, and its orientation is used only for choosing the appropriate position of the chain (c). Note that the first series of Borromean links with bigons could also be derived from Borromean rings by introducing identical even chains of bigons in the crossing-points of two different components. Therefore, we could first get different infinite series of n-Borromean links without bigons, and then introduce bigons in a way which preserves the Borromean property.