Summary: CIRCLE LINKS
The Borromean rings have the property that removing one of the components leaves the unlink.
Since each component is unknotted, and an unknot may be represented by a round circle, can one
make the Borromean rings out of round circles? It is not too hard to show that the Borromean rings
may be made out of ellipses which are arbitrarily close to circles. To see this, take three orthogonal
planes, and three congruent non-circular ellipses lying in the orthogonal planes with centers at the
intersection, such that the major and minor axes of the ellipses lie in the intersections between pairs
of planes (see figure 1).
Figure 1. Ellipse Borromean rings
If one chooses these so that the ellipses are disjoint, then the union of the three ellipses form the
Borromean rings. Letting the major and minor radii approach each other, one obtains Borromean
rings with the components arbitrarily close to circles. Michael Freedman has shown that one cannot
make the components simultaneously circular. More generally, if one has a link for which every pair
of components have linking number 0, then the link is the unlink: it may be isotoped so that each
component is a circle which bounds a disk disjoint from all the other components. This proof is an
example of a "book proof", a proof which makes the theorem transparent. Erd¨os believed that God
has a book with the most elegant proofs of every theorem written in it, and one of mathematicians'
goals is to discover what is written in this book.
Freedman's proof uses the fact that R3
is a subspace of R4