## Topology for ComputingThe emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate. |

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### Contents

1 | |

V | 3 |

VI | 8 |

VII | 10 |

VIII | 11 |

IX | 13 |

X | 14 |

XI | 19 |

XXXIX | 148 |

XL | 150 |

XLI | 153 |

XLII | 157 |

XLIII | 161 |

XLIV | 162 |

XLVI | 166 |

XLVII | 169 |

XII | 23 |

XIII | 32 |

XIV | 37 |

XV | 41 |

XVII | 47 |

XVIII | 53 |

XIX | 60 |

XXI | 70 |

XXII | 79 |

XXIII | 83 |

XXIV | 84 |

XXV | 85 |

XXVI | 86 |

XXVII | 88 |

XXVIII | 90 |

XXIX | 94 |

XXX | 95 |

XXXI | 105 |

XXXII | 116 |

XXXIII | 123 |

XXXIV | 125 |

XXXVI | 128 |

XXXVII | 136 |

XXXVIII | 146 |

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### Common terms and phrases

2-manifold algebra algorithm algorithm for computing basis Betti numbers binary operation boundary canonical cycles Chapter coefficients column combinatorial component computing persistence connected corresponding cosets critical points CView cyclic groups data set data structure defined Definition diagram dimension dimensional Edelsbrunner edge elements eliminate equivalence Euler characteristic example field filtration free Abelian group functor fundamental group geometry Gramicidin graph grid group G homeomorphic homology groups homotopy implementation integer integral domain intersection isomorphic knot l-cycles linking number loop metric Morse complex Morse function Morse theory Morse-Smale complex neighborhood nonbounding oriented P-intervals p-linked pairs path persistence algorithm persistence module persistent complexes positive simplices protein QMS complex quadrangles region reordering algorithms ring saddle Section Seifert surface shown in Figure simplicial complex spanning surface sphere subgroup subset Table tangent terrain Theorem topological space topology map torsion torus triangle union-find unstable manifolds vector space vertex vertices visualization zeolite

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