# Computational Topology: An Introduction

American Mathematical Soc., 2010 - Mathematics - 241 pages
Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

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### Contents

 Graphs 3 12 Curves in the Plane 9 13 Knots and Links 13 14 Planar Graphs 18 Exercises 24 Surfaces 27 II2 Searching a Triangulation 33 II3 Selfintersections 37
 Exercises 123 Morse Functions 125 VI2 Transversality 130 VI3 Piecewise Linear Functions 135 VI4 Reeb Graphs 140 Exercises 145 Computational Persistent Topology 147 Persistence 149

 II4 Surface Simplification 42 Exercises 47 Complexes 51 III2 Convex Set Systems 57 III3 Delaunay Complexes 63 III4 Alpha Complexes 68 Exercises 74 Computational Algebraic Topology 77 Homology 79 IV2 Matrix Reduction 85 IV3 Relative Homology 90 IV4 Exact Sequences 95 Exercises 101 Duality 103 V2 Poincaré Duality 108 V3 Intersection Theory 114 V4 Alexander Duality 118
 VII2 Efficient Implementations 156 VII3 Extended Persistence 161 VII4 Spectral Sequences 166 Exercises 171 Stability 175 VIII2 Stability Theorems 180 VIII3 Length of a Curve 185 VIII4 Bipartite Graph Matching 191 Exercises 197 Applications 199 IX2 Elevation for Protein Docking 206 IX3 Persistence for Image Segmentation 213 IX4 Homology for Root Architectures 218 Exercises 224 References 227 Index 235 Copyright

### About the author (2010)

Herbert Edelsbrunner is Arts and Sciences Professor of Computer Science at Duke University. He was the winner of the 1991 Waterman award from the National Science Foundation and is the founder and director of Raindrop Geomagic, a 3-D modelling company.