Beginning first with my thesis at Brown University, then with work done at the Geometry Center at the University of Minnesota, and continuing here at Union College, I have studied polyhedral immersions of surfaces that have certain combinatorial or geometric properties. The combinatorial condition is that of having the minimum number of vertices that permit an immersion in three-space; the geometric one, that of having a slicing property called tightness, which is closely related to convexity.
A combinatorial argument gives a lower bound on the number of vertices necessary to produce a triangulation of a surface, with this bound depending only on the Euler characteristic (a topological invariant) of the surface. For example, the minimum number for a torus is seven. In 1949,
Császár found a polyhedral embedding (that is, a one-to-one mapping) of a seven-vertex torus into three-space, the first non-trivial example of an embedding of a vertex-minimal triangulation into space. More recently, Brehm ,  and Bokowski and Brehm have constructed such mappings for surfaces of higher genus.
Non-orientable surfaces can not be embedded in three-space, but we can look for mappings that are locally embedded; that is, where the disk neighborhood of every vertex is mapped one-to-one. Such a mapping is called an immersion. For example, the real projective plane can be triangulated using only six vertices, but
Brehm showed that it requires nine vertices to immerse this surface in three-space. This follows directly from a theorem of Banchoff that guarantees the existence of a triple point in any immersion of a surface with odd Euler characteristic; such a triple point must be formed by three intersecting triangles that have no vertices in common.
The most natural surface to consider next is the Klein bottle. It requires eight vertices for a triangulation, and the obvious question is: Can the Klein bottle be immersed in space with only eight vertices? In my
thesis , I produced several distinct nine-vertex immersions, and was able to show that no eight-vertex immersion exists. This is the first result about vertex-minimal immersions that does not follow directly either from an embedding of a minimal triangulation or from Banchoff's theorem.
Views of two distinct nine-vertex Klein bottles. The one on the left is in the standard immersion class with reflective symmetry; the one on the right is in a non-standard class with rotational symmetry.
Little is known about vertex-minimal immersions of other non-orientable surfaces, or of orientable surfaces with higher genus, so there is quite a bit of work still to do in this area. For example, it is likely that as the genus increases, the number of vertices required for an embedding of an orientable surface rises faster than the number required for a triangulation of the surface; what can be said about the difference in rates? For surfaces through genus four, we know that the vertex-minimal embedding is of a vertex-minimal triangulation; what is the smallest genus where the embedding requires more vertices than a minimal triangulation? Is there a genus for which there is an immersion with fewer vertices than any embedding of that surface?
A tight polyhedral immersion of the real projective plane with one handle.
A surface is said to be tight provided it has the
two-piece property ; namely, that any plane cuts the surface into at most two pieces. Convex surfaces have this property, but some non-convex surfaces do as well, for example a torus of revolution. The surface of a banana, however, is not tight since a single plane can cut off both ends and thereby separate the surface into three pieces.
A natural question is: What surfaces can be tightly immersed in three-space?
Kuiper [33–35]answered this question for all surfaces except for one: the projective plane with one handle. The result for this last surface remained unknown for over thirty years, until in 1992, Haab showed that no smooth tight immersion is possible. It came as a surprise, then, when I showed two years later [19,18]that there is a polyhedral tight immersion. This represents one of only a handful of low-dimensional examples where the smooth and polyhedral theories differ in a significant way.
We can refine the question further and ask: For those surfaces for which tight immersions are possible, how many different such immersions are there? Pinkall
showed that the various types of tight immersions fall into ten distinct, but infinite, families, and he gave smooth and polyhedral examples of all the possible classes except for a finite number of them with low genus. In , I develop polyhedral versions of all but three of the remaining classes, and point out and correct an error in Pinkall's work that would have left an infinite family with no known tight immersions. Subsequently, I produced examples of two of the missing three classes and conjecture that the final one can not be tightly immersed. Haab has proven this conjecture in the smooth case, but it remains an open question for polyhedra.
There are several outstanding questions remaining to be addressed. First, why is there a tight polyhedral immersion of the real projective plane but no smooth one? That is, what is the obstruction to smoothing the tight model developed
in ? In I begin to address this question, showing that there is no local impediment, but a complete understanding of why the polyhedral model can not be smoothed while retaining tightness remains elusive. Another question is how unique is the surface in ? Is there a family of related tight immersions, for example?
In his paper, Pinkall generated smooth examples from his polyhedral ones using a smoothing algorithm; my models, however, do not satisfy the requirements for his process, so the results
in  and do not necessarily carry over to the smooth case. It may well be that some of these also represent examples of surfaces like the projective plane with one handle, where a polyhedral, but no smooth, tight immersion exists. The smoothability of my models remains to be investigated. Finally, there is still the class for which I have conjectured that no polyhedral tight immersion exists; a proof of this seems quite difficult.
The projective plane with one handle has another unique property, which I describe
in , namely, that it has no tight immersion in three-space that is symmetric. Every other surface that has a tight immersion has a symmetric one (and usually several different symmetries can be obtained). The implications of this essential asymmetry are not yet clear, but it does suggest other lines of investigation. For example, can the symmetries possible for other surfaces be classified? I have developed a number of conditions under which various symmetries can be guaranteed. As might be expected, the genus of the surface plays an important role in determining what rotational symmetries are possible, and it is not hard to show that, for any n, there are only a finite number of surfaces for which an n-fold rotational symmetry is not possible. One could extend this study to look at the various homotopy classes of surfaces described by Pinkall in .
In addition to these questions in three-space, there are open problems dealing with tight surfaces in higher dimensions as well. For example, no smooth tight immersion of the Klein bottle in four-space is known, though there is a polyhedral one; can one be found? Does the real projective plane with one handle have symmetric tight immersions in four-, five- or six-space (the only dimensions in which it can be substantially embedded)? I conjecture that the answer is "yes" in all three cases. Vertex-minimal immersions of higher-dimensional manifolds may also yield some interesting results; for instance, Hughes
conjectures that there is a 16-vertex immersion of the three-sphere in four-space whose cross sections form an eversion of the sphere.
Recently, I have begun new work on the classification of non-standard polyhedral saddle points and of their possible combinations in embedded and immersed surfaces. Unlike the smooth case, where there is only one saddle for of each index, polyhedral saddles come in various forms. For example, there are two index 2 saddle types: the standard "monkey saddle", and an exotic saddle that can not be smoothed without breaking it up into two saddles of index 1. For index 3, there are five combinatorially different saddles, and the number goes up rapidly from there. Little is known about the exotic saddles for index even as low as 8. Preliminary work with Tom Banchoff and Ockle Johnson may lead to a joint paper investigating a combinatorial approach to measuring the Stiffle-Whitney class for surfaces in four-space. The
StageToolssoftware, discussed below, is proving to be an important tool in developing the ideas necessary to tackle this problem.