An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces: 1st Edition (Hardback) book cover

An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces

1st Edition

By David Jackson, Terry I. Visentin

Chapman and Hall/CRC

296 pages

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Hardback: 9781584882077
pub: 2000-09-15
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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.

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

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COM046000
COMPUTERS / Operating Systems / General
MAT012000
MATHEMATICS / Geometry / General
MAT036000
MATHEMATICS / Combinatorics