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This book presents a topological approach to combinatorial configurations, in particular graphs, by introducing a new pair of homology and cohomology via polyhedra. On this basis, a number of problems are solved using a new approach, such as the embeddability of a graph on a surface (orientable and nonorientable) with given genus, the Gauss crossing conjecture, the graphicness and cographicness of a matroid, and so forth. Notably, the specific case of embeddability on a surface of genus zero leads to a number of corollaries, including the theorems of Lefschetz (on double coverings), of MacLane (on cycle bases), and of Whitney (on duality) for planarity. Relevant problems include the Jordan axiom in polyhedral forms, efficient methods for extremality and for recognizing a variety of embeddings (including rectilinear layouts in VLSI), and pan-polynomials, including those of Jones, Kauffman (on knots), and Tutte (on graphs), among others.
Contents
Homology on Polyhedra
Polyhedra on the Sphere
Automorphisms of a Polyhedron
Gauss Crossing Sequences
Cohomology on Graphs
Embeddability on Surfaces
Embeddings on Sphere
Orthogonality on Surfaces
Net Embeddings
Extremality on Surfaces
Matroidal Graphicness
Knot Polynomials
Table of Content: Preface Chapter 1 Preliminaries 1.1 Sets and relations 1.2 Partitions and permutations 1.3 Graphs and networks 1.4 Groups and spaces 1.5 Notes Chapter 2 Polyhedra 2.1 Polygon double covers 2.2 Supports and skeletons 2.3 Orientable polyhedra 2.4 Nonorientable polyhedral 2.5 Classic polyhedral 2.6 Notes Chapter 3 Surfaces 3.1 Polyhegons 3.2 Surface closed curve axiom 3.3 Topological transformations 3.4 Complete invariants 3.5 Graphs on surfaces 3.6 Up-embeddability 3.7 Notes Chapter 4 Homology on Polyhedra 4.1 Double cover by travels 4.2 Homology 4.3 Cohomology 4.4 Bicycles 4.5 Notes Chapter 5 Polyhedra on the Sphere 5.1 Planar polyhedra 5.2 Jordan closed curve axiom 5.3 Uniqueness 5.4 Straight line representations 5.5 Convex representation 5.6 Notes Chapter 6 Automorphisms of a Polyhedron 6.1 Automorphisms 6.2 V -codes and F-codes 6.3 Determination of automorphisms 6.4 Asymmetrization 6.5 Notes Chapter 7 Gauss Crossing Sequences 7.1 Crossing polyhegons 7.2 Dehns transformation 7.3 Algebraic principles 7.4 Gauss Crossing problem 7.5 Notes Chapter 8 Cohomology on Graphs 8.1 Immersions 8.2 Realization of planarity 8.3 Reductions 8.4 Planarity auxiliary graphs 8.5 Basic conclusions 8.6 Notes Chapter 9 Embeddability on Surfaces 9.1 Joint tree model 9.2 Associate polyhegons 9.3 A transformation 9.4 Criteria of embeddability 9.5 Notes Chapter 10 Embeddings on the Sphere 10.1 Left and right determinations 10.2 Forbidden Congurations 10.3 Basic order characterization 10.4 Number of planar embeddings 10.5 Notes Chapter 11 Orthogonality on Surfaces 11.1 Denitions 11.2 On surfaces of genus zero 11.3 Surface Model 11.4 On surfaces of genus not zero 11.5 Notes Chapter 12 Net Embeddings 12.1 Denitions 12.2 Face admissibility 12.3 General criterion 12.4 Special criteria 12.5 Notes Chapter 13 Extremality on Surfaces 13.1 Maximal genus 13.2 Minimal genus 13.3 Shortest embedding 13.4 Thickness 13.5 Crossing number 13.6 Minimal bend 13.7 Minimal area 13.8 Notes Chapter 14 Matroidal Graphicness 14.1 Denitions 14.2 Binary matroids 14.3 Regularity 14.4 Graphicness 14.5 Cographicness 14.6 Notes Chapter 15 Knot Polynomials 15.1 Denitions 15.2 Knot diagram 15.3 Tutte polynomial 15.4 Pan-polynomial 15.5 Jones polynomial 15.6 Notes Reference Index
Yanpei Liu , Beijing Jiaotong University, Beijing, China.
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