Toroidal Triangulations with Flat Faces 

This problem concerns triangulations of the torus T and their representations in three-dimensional Euclidean space R^3. Specifically, a homeomorph of a triangulation of the torus in 3-space is flat if every face is contained in a 2-dimensional subspace. Note that if a triangulation has a flat toroidal homeomorph, then it must be simple. The word ``flat'' has been used in another context for embeddings of graphs in 3-space, however, there should be no confusion here as we are talking of flat torii in 3-space.

The following conjecture is due to Richard Duke.

Conjecture: Every simple toroidal triangulation has a flat homeomorph in R^3.

I know of no partial results towards this conjecture. Lawrenchenko has found all minor-minimal triangulations of the torus. However, it is not clear to me if having a flat homeomorph is preserved under minors. If it is, then the above conjecture reduces the conjecture to checking a small number of cases. Lawrenchenko mentioned to me that he has a possible approach to the problem; the interested reader should consult him.

Terry McKee has a nice flat homeomorph of K_7 on the torus. The model has the curious property that some faces are quite small in relation to others despite the symmetry of the embedding. I believe that he mentioned a result stating that this imbalance must occur in any flat homeomorph of this embedding.

Submitted by: Dan Archdeacon, Department of Mathematics and Statistics, University of Vermont, Burlington VT 05405 USA.

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August, 1995