The Key-Ring Model of the Hopf Fibration

The 3-dimensional sphere S3 serves as a fundamental example in several areas of geometry and topology. The simplest way to define it, perhaps, is to say that it is the set of unit vectors in 4-dimensional space:

S3 = { (w,x,y,z): w2+x2+y2+z2=1 }.
One of the amazing things about the 3-sphere is that it has a Hopf fibration. This is a partition of S3 into a differentiable collection of circles in such a way that every pair of circles is linked and the quotient topology yields the "ordinary" sphere S2 in 3-dimensional space. Despite the fact that this object exists naturally in four dimensions, one can build an incredibly simple and intuitive 3-dimensional model. Merely assemble a collection of "split" key rings so that each pair of rings is linked:

Key-Ring Model of the Hopf Fibration

The model pictured has 50 steel split key rings, each with a diameter of 2.25 in (5.7 cm), and, of course, assembled so that each pair of rings constitutes a Hopf link. Another defining characteristic of the model is that each triple of key rings has the same topological orientation, as an imbedding in 3-space, as every other such triple. Of course, since the Hopf fibration is comprised of a continuum of infinitely many key rings, the model represents merely an approximation of the Hopf fibration. One may also argue that it is not a particularly faithful model because it does not respect the metrical properties of stereographic projection. Nevertheless, it is a remarkably simple and intuitive model, faithful to several of the critical properties of the Hopf fibration of the 3-sphere.

Notice finally that one may build two distinct orientations:

Both Orientations of the Key-Ring Model

One way to interpret the existence of these distinct orientations goes as follows. The 3-sphere is a Lie group and the circle is a closed subgroup. The Hopf fibration corresponds to the partition of S3 into cosets by left- or right-orbits of the circle subgroup. Since the group S3 is non-abelian, the partitions into left cosets and right cosets are distinct, thus yielding the two distinct orientations as pictured.

January 23, 2005.