Minimal Fibrations of Hyperbolic 3-manifolds

There are hyperbolic 3-manifolds that fiber over the circle but that do not admit fibrations by minimal surfaces. These manifolds do not admit fibrations by surfaces that are even approximately minimal

Invariants of Knot Diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams

Double Bubbles Minimize

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as described in the article. You can obtain this code by viewing the source of this articl

A Metric for genus-zero surfaces

We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms give a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.Comment: 33 pages, 8 figure

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry

Configurations of curves and geodesics on surfaces

We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.Comment: 13 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper11.abs.htm

The Number of Triangles Needed to Span a Polygon Embedded in R^d

Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface construction to show there always exists an embedded surface requiring at most 7n^2 triangles. We complement this result by showing there are polygons in R^3 for which any embedded surface requires at least 1/2n^2 - O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions 5 or more there exists an embedded surface requiring at most n triangles. In dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded surfaces, and a construction of an immersed disk requiring at most 3n triangles. These results can be interpreted as giving qualitiative discrete analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of math.GT/020217