This monograph is for a unified theory of surfaces, embeddings and maps all considered as polyhedra via the joint tree modal which was initiated from the author's articles in the seventies of last century and has been basically developed in recent decades. Complete invariants for each classification are topologically, combinatorially or isomorphically extracted. A number of counting polynomials including handle and crosscap polynomials are presented. In particular, an appendix serves as the exhaustive counting super maps (rooted and nonrooted) including these polynomials with under graphs of small size for the reader's digests.
Although the book is mainly for researchers in mathematics, theoretical physics, chemistry, biology and some others related, the basic part in each chapter can also be chosen for graduates and college teachers as references.
Preface
Chapter I Preliminaries
I.1 Sets and mappings
I.2 Partitions and permutations
I.3 Group actions
I.4 Networks
I.5 Notes
Chapter II Surfaces
II.1 Polyhedra
II.2 Elementary equivalence
II.3 Polyhegons
II.4 Orientability
II.5 Classification
II.6 Notes
Chapter III Embeddlngs of Graphs
III.1 Geometric consideration
III.2 Surface closed curve axiom
III.3 Distinction
III.4 Joint tree model
III.5 Combinatorial properties
III.6 Notes
Chapter IV Mathematical Maps
IV.1 Basic permutations
IV.2 Conjugate axiom
IV.3 Transitivity
IV.4 Included angles
IV.5 Notes
Chapter V Duality on Surfaces
V.1 Dual partition of edges
V.2 General operation
V.3 Basic operations
V.4 Quadrangulations
V.5 Notes
Chapter VI Invariants on Basic Class
VI.1 Orientability
VI.2 Euler characteristic
VI.3 Basic equivalence
VI.4 Orientable maps
VI.5 Nonorientable maps
VI.6 Notes
Chapter VII Asymmetrization
VII.1 Isomorphisms
VII.2 Recognition
VII.3 Upper bound of group order
VII.4 Determination of the group
VII.5 Rootings
VII.6 Notes
Chapter VIII Asymmetrized Census
VIII.1 Orientable equation
VIII.2 Planar maps
VIII.3 Nonorientable equation
VIII.4 Gross equation
VIII.5 The number of maps
VIII.6 Notes
Chapter IX Petal Bundles
IX.1 Orientable petal bundles
IX.2 Planar pedal bundles
IX.3 Nonorientable pedal bundles
IX.4 The number of pedal bundles
IX.5 Notes
Chapter X Super Maps of Genus Zero
X.1 Planted trees
X.2 Outerplanar graphs
X.3 Hamiltonian planar graphs
X.4 Halin graphs
X.5 Notes
Chapter XI Symmetric Census
XI.1 Symmetric relation
XI.2 An application
XI.3 Symmetric principle
XI.4 General examples
XI.5 Notes
Chapter XII Cycle Oriented Maps
XII.1 Cycle orientation
XII.2 Pan-bouquets on surfaces
XII.3 Boundary maps
XII.4 Graphs on surfaces
XII.5 Notes
Chapter XIII Census by Genus
XIII.1 Associate surfaces
XIII.2 Layer division of a surface
XIII.3 Handle polynomials
XIII.4 Crosscap polynomials
XIII.5 Maps from embeddings
XIII.6 Graphs with same polynomial
XIII.7 Notes
Chapter XIV Classic Applications
XIV.1 Convex embeddings
XIV.2 Rectilinear embeddings
XIV.3 Boundary thickness
XIV.4 Dehn diagram on knots
XIV.5 Potts models in theoretical physics
XIV.6 Notes
Appendix I Embeddings and Maps of Small Size Distributed by Genus
Ax.I.1 Triconnected cubic graphs
Ax.I.2 Bouquets
Ax.I.3 Wheels
Ax.I.4 Link bundles
Ax.I.5 Complete bipartite graphs
Ax.I.6 Quadregular graphs
Appendix II Orientable Forms of Surfaces and Their Non- orientable Genus Polynomials
Ax.II.1 Forms of orientable 2β-surfaces
Ax.II.2 Nonorientable genus polynomials
Bibliography
Subject Index
Author Index
