Table of Contents
1 Preliminaries 1
1.1 Sets and relations 1
1.2 Partitions and permutations 5
1.3 Graphs and networks 9
1.4 Groups and spaces 15
1.5 Notes 20
2 Polyhedra 22
2.1 Polygon double covers 22
2.2 Supports and skeletons 25
2.3 Orientable polyhedra 28
2.4 Non-orientable polyhedra 30
2.5 Classic polyhedra 32
2.6 Notes 34
3 Surfaces 35
3.1 Polyhegons 35
3.2 Surface closed curve axiom 39
3.3 Topological transformations 42
3.4 Complete invariants 46
3.5 Graphs on surfaces 48
3.6 Up-embeddability 52
3.7 Notes 56
4 Homology on Polyhedra 58
4.1 Double cover by travels 58
4.2 Homology 60
4.3 Cohomology 65
4.4 Bicycles 70
4.5 Notes 75
5 Polyhedra on the Sphere 78
5.1 Planar polyhedra 78
5.2 Jordan closed-curve axiom 84
5.3 Uniqueness 87
5.4 Straight-line representations 91
5.5 Convex representation 93
5.6 Notes 95
6 Automorphisms of a Polyhedron 97
6.1 Automorphisms of polyhedra 97
6.2 Eulerian and non-Eulerian codes 102
6.3 Determination of automorphisms 110
6.4 Asymmetrization 124
6.5 Notes 127
7 Gauss Crossing Sequences 129
7.1 Crossing polyhegons 129
7.2 Dehn's transformation 133
7.3 Algebraic principles 137
7.4 Gauss crossing problem 141
7.5 Notes 143
8 Cohomology on Graphs 145
8.1 Immersions 145
8.2 Realization of planarity 148
8.3 Reductions 151
8.4 Planarity auxiliary graphs 154
8.5 Basic conclusions 158
8.6 Notes 163
9 Embeddability on Surfaces 165
9.1 Joint tree model 165
9.2 Associate polybegons 167
9.3 A transformation 168
9.4 Criteria of embeddability 171
9.5 Notes 173
10 Embeddings on Sphere 175
10.1 Left and right determinations 175
10.2 Forbidden configurations 179
10.3 Basic order characterization 185
10.4 Number of planar embeddings 192
10.5 Notes 197
11 Orthogonality on Surfaces 198
11.1 Definitions 198
11.2 On surfaces of genus zero 205
11.3 Surface models 225
11.4 On surfaces of genus not zero 228
11.5 Notes 229
12 Net Embeddings 231
12.1 Definitions 231
12.2 Face admissibility 236
12.3 General criterion 242
12.4 Special criteria 248
12.5 Notes 255
13 Extremality on Surfaces 257
13.1 Maximal genus 257
13.2 Minimal genus 261
13.3 Shortest embedding 264
13.4 Thickness 273
13.5 Crossing number 275
13.6 Minimal bend 277
13.7 Minimal area 283
13.8 Notes 288
14 Matroidal Graphicness 291
14.1 Definitions 291
14.2 Binary matroids 292
14.3 Regularity 295
14.4 Graphicness 300
14.5 Cographicness 305
14.6 Notes 306
15 Knot Polynomials 308
15.1 Definitions 308
15.2 Knot diagram 312
15.3 Tutte polynomial 317
15.4 Pan-polynomial 320
15.5 Jones Polynomial 327
15.6 Notes 329
Bibliography 331
Subject Index 347
Author Index 355
