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An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces

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Overview

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.

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Product Details

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
INDEX

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