Restricting valence for polyhedral surfaces and manifolds
Project leader: J. M. Sullivan
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
- 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.