Project S

Restricting valence for polyhedral surfaces and manifolds

This is a continuation of the previous project Discrete minimal surfaces in the cubic lattice.

The construction of a combinatorial manifold, from simplices or other polyhedral facets, is largely determined by the valence - how many facets fit around each face of codimension two. Valence, and the restrictions one can place on it, is a common theme uniting the several parts of this project.

The work on this project during the first funding period, as well as our planned work during the next period, can be divided into several parts:

• Simplicial manifolds of small valence: Every 3-manifold can be built using valence 4,5,6, but our work in the first period showed that only a few 3-manifolds (all with spherical geometry) can be built with valence at most 5. We will investigate whether valence 5 and 6 suffice, hopefully leading to a classification of euclidean and noneuclidean TCP triangulations.
• Quadrangulations with high valence and equivelar surfaces: For any surface with χ<0, the triangulations of highest average valence are those with the minimum number of triangles; Ringel and Jungerman showed that an elementary upper bound for the highest average valence is sharp in almost all cases. We will investigate the analogous problem for quadrangulations, aiming to get a sharp bound. We will also enumerate equivelar surfaces tiled by p-gons.
• Higher-dimensional soap films: Possible singularities for soap bubbles in higher dimensions are related to generalized deltatopes, i.e. convex polytopes in the appropriate dimension whose 3-faces are euclidean regular tetrahedra. This geometry implies that such polytopes necessarily have low valence. We will enumerate the generalized deltatopes, which include a large subfamily related to the E_8 lattice. We will then test the resulting candidate singularities, to see if they actually can occur in higher-dimensional soap films.
• Realizability in a given isotopy class: We will investigate the realizability of polyhedral surfaces. In particular, we will consider the effect of requiring that the linear embedding in R^3 belongs to a particular isotopy class.