## Computational Topology: An IntroductionCombining 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

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 |

227 | |

235 | |

### Common terms and phrases

2-manifold 3-dimensional abstract simplicial complex algorithm alpha complex assume Betti numbers Bibliographic notes boundary matrix chain complex circle closed curve cohomology column components compute connected consider construction continuous map convex corresponding credits critical points cycle defined diagonal dimension disjoint disk distance dual block edges embedding endpoints equivalent Euler characteristic exact sequence example Figure finite geometric height function homology class homomorphism homotopy type Hp(K implies intersection isomorphic Klein bottle Lefschetz Duality Lemma line segment loops lower star manifold Mayer-Vietoris sequence minimum Morse function node non-empty p-chain p-th pair path persistence diagrams persistent homology piecewise linear planar graph plane Poincaré duality polygon proof radius rank reduced homology Reeb graph relative homology groups saddle simplex simplices simplicial complex sphere stable manifolds subcomplex sublevel set Theorem topological space torus triangles vector spaces vertex vertices Voronoi zero