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

On limits of Graphs Sphere Packed in Euclidean Space and Applications

The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be extended to higher dimensions. In particular, it is shown that every limit of finite graphs sphere packed in $\R^d$ with a uniformly-chosen root is $d$-parabolic. We then derive few geometric corollaries. E.g.\,every infinite graph packed in $\R^{d}$ has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets $W$ with boundary size which satisfies $|\partial W| \leq |W|^{\frac{d-1}{d}+o(1)}$. Some open problems and conjectures are gathered at the end

Topological Phases: An Expedition off Lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of `baby universe', Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger's theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.Comment: 38 pages, 22 figure