Computational Topology by Edelsbrunner and Harer

Contents

Part a

• Graphs 3
  • Curves in the plane 9
  • Knots and links 13
  • Planar graphs 18

• Surfaces 27
  • Searching a triangulation 33
  • Selfintersections 37
  • Surface simplification 42

• Complexes 51
  • Convex set systems 57
  • Delaunay complexes 63
  • Alpha complexes 68

Part b: Computational algebraic topology 77

• Homology 79
  • Matrix reduction 85
  • Relative homology 90
  • Exact sequences 95

• Duality 103
  • Poincare duality 108
  • Intersection theory 114
  • Alexander duality 118

• Morse functions 125
  • Transversality 130
  • Piecewise linear functions 135
  • Reeb graphs 140

Part b: Computational persistent topology 147

• Persistence 149
  • Efficient implementations 156
  • Extended persistence 161
  • Spectral sequences 166

• Stability 175
  • Stability theorems 180
  • lengths of curves 185
  • Bipartite graph matching 191

• Applications 199
  • Elevation for protein docking 206
  • Persistence for image segmentation 213
  • Homology for root architectures 218