An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces

An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces

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Description

Maps are beguilingly simple structures with deep and ubiquitous properties. They arise in an essential way in many areas of mathematics and mathematical physics, but require considerable time and computational effort to generate. Few collected drawings are available for reference, and little has been written, in book form, about their enumerative aspects. An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces is the first book to provide complete collections of maps along with their vertex and face partitions, number of rootings, and an index number for cross referencing. It provides an explanation of axiomatization and encoding, and serves as an introduction to maps as a combinatorial structure. The Atlas lists the maps first by genus and number of edges, and gives the embeddings of all graphs with at most five edges in orientable surfaces, thus presenting the genus distribution for each graph. Exemplifying the use of the Atlas, the authors explore two substantial conjectures with origins in mathematical physics and geometry: the Quadrangulation Conjecture and the b-Conjecture. The authors' clear, readable exposition and overview of enumerative theory makes this collection accessible even to professionals who are not specialists. For researchers and students working with maps, the Atlas provides a ready source of data for testing conjectures and exploring the algorithmic and algebraic properties of maps.show more

Product details

  • Hardback | 296 pages
  • 162.6 x 242.8 x 21.6mm | 636.99g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 96 black & white tables
  • 1584882077
  • 9781584882077

Table of contents

PREFACE MAPS INTRODUCTION Organization of the Atlas Further Reading SURFACES AND MAPS Representation of Maps and Surfaces Examples of the Definitions Using the Atlas An Application of k-Realizable Partitions THE AXIOMATIZATION AND THE ENCODING OF MAPS Orientable Surfaces Locally Orientable Surfaces GENERATING SERIES AND CONJECTURES Generating Series for Hypermaps Specialization to Maps The Quadrangulation Conjecture The b-Conjecture THE ATLAS MAPS IN ORIENTABLE SURFACES Genus 0 - The Sphere Genus 1 - The Torus Genus 2 - The Double Torus MAPS IN NONORIENTABLE SURFACES Genus 1 - The Projective Plane Genus 2 - The Klein Bottle Genus 3 - The Crosscapped Torus Genus 4 - The Doubly Crosscapped Torus FACE REGULAR MAPS AND HYPERMAPS Triangulations Quadrangulations Hypermaps ASSOCIATED GRAPHS AND THEIR MAPS TABLES NUMBERS OF ROOTED MAPS Orientable: by Vertex and Face Partition Nonorientable: by Vertex and Face Partition Summarized by Edges and Vertices NUMBERS OF UNROOTED MAPS Orientable: by Vertex and Face Partition Nonorientable: by Vertex and Face Partition NONREALIZABLE PAIRS OF PARTITIONS For Orientable Surfaces For Nonorientable Surfaces MAP POLYNOMIALS b-Polynomials Genus Distributions BIBLIOGRAPHY GLOSSARY INDEXshow more