Compression of orientable 3-manifold triangulations and Pachner paths
Abstract
Three dimensional triangulations can be described by giving a set of gluing maps between faces of tetrahedra (subject to some mild constraints). While this is a natural way to describe triangulations, it becomes computationally expensive to recognize when two triangulations are isomorphic. Here isomorphic triangulations are equivalent up to relabelling. To solve this problem, Burton created an isomorphism signature, which associates a string canonically to a triangulation that is shared by all triangulations isomorphic to it. However, this representative labelling never corresponds to an oriented triangulation. In computational topology, it is often important to deal with oriented triangulations if possible so we present a similar encoding for orientable 3-manifolds known as an oriented isomorphism signature that will always encode an oriented triangulation. We also present an encoding scheme for describing a path in the Pachner graph, an object for relating all triangulations of a fixed 3-manifold, as a string of printable characters that can be appended to the end of an oriented isomorphism signature. This allows us to easily store and describe how one isomorphism class of triangulations can be transformed into another via a series of local operations without losing any topological data.
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- OSU Theses [15752]