Linear algebraic representation for topological structures

Abstract

With increased complexity of geometric data, topological models play an increasingly important role beyond boundary representations, assemblies, finite elements, image processing, and other traditional modeling applications. While many graph- and index-based data structures have been proposed, no standard representation has emerged as of now. Furthermore, such representations typically do not deal with representations of mappings and functions and do not scale to support parallel processing, open source, and client-based architectures. We advocate that a proper mathematical model for all topological structures is a (co)chain complex: a sequence of (co)chain spaces and (co)boundary mappings. This in turn implies all topological structures may be represented by a collection of sparse matrices. We propose a Linear Algebraic Representation (LAR) scheme for mod 2 (co)chain complexes using CSR matrices and show that it supports a variety of topological computations using standard matrix algebra, without any overhead in space or running time. A full open source implementation of LAR is available and is being used for a variety of applications.