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.
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目录
Preface Chapter I Preliminaries I.1 Sets and mappings I.2 Partitions and permutations I.3 Group action 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 Embeddings 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.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 Nonorientable Genus Polynomials Ax.II.1 Forms of orientable 2β-surfaces Ax.II.2 Nonorientable genus polynomials Bibliography Subject Index Author Index
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