MINIMAL TRIANGULATIONS AND NORMAL SURFACES

This thesis examines three distinct problems relating to the combinatorial structures of minimal 3-manifold triangulations and to the study of normal surfaces within these triangulations. These problems include the formation and analysis of a census of 3-manifold triangulations, a study of splitting surfaces within 3-manifold triangulations and an investigation into the complexity of the normal surface solution space. An algorithm for generating a census of all closed prime minimal 3-manifold triangulations is presented, extending the algorithms of earlier authors in several ways. Automorphisms of face pairings are utilised to improve the efficiency of the generation of triangulations. 0-efficiency tests and searches for particular subcomplexes within a triangulation are introduced to aid the subsequent processing of these triangulations. Results involving face pairing graphs are proven for the purpose of eliminating large classes of triangulations at different stages of the algorithm. Using this algorithm, a census is formed of all closed prime minimal triangulations containing at most six tetrahedra. The census of non-orientable triangulations in particular is the first such census to be published. A detailed analysis is performed of the underlying combinatorial structures of the resulting triangulations, extending current knowledge in both the orientable and non-orientable cases. Also included is a full listing of the vertex normal surfaces of these triangulations. An infrastructure is then developed for studying splitting surfaces within 3-manifold triangulations. Splitting surfaces represent a particular class of normal surfaces containing only quadrilateral discs, and have several interesting combinatorial and topological properties. Splitting surface signatures are introduced to assist with representation and computation, and a census of all splitting surface signatures of order ≤ 8 is presented. The final problem, which explores the complexity of the normal surface solution space, concentrates on bounding the number of maximal embedded faces of the projective solution space. An extensive analysis is performed upon the geometric structure of this solution space when represented using edge weight coordinates. In the case of general position this geometric structure is used to formulate a corresponding algebraic problem from which partial results are obtained. Both the study of the census algorithm and the analysis of the resulting minimal triangulations draw upon previously published results. These results are enhanced and significant new related results are offered. In addition, the computer software written to support this work provides a powerful tool for the study of normal surfaces and 3-manifold triangulations. The investigations into splitting surfaces and the complexity of the normal surface solution space are entirely new. The theory that is developed constitutes a strong basis for continued research into these areas.

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