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P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 This page intentionally left blank vi P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 CAMBRIDGEMONOGRAPHSON APPLIEDANDCOMPUTATIONAL MATHEMATICS SeriesEditors P.G.CIARLET,A.ISERLES,R.V.KOHN,M.H.WRIGHT 16 Topology for Computing i P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 TheCambr¨ıdgeMonographsonAppliedandComputationalMathematicsreflectsthe crucialroleofmathematicalandcomputationaltechniquesincontemporaryscience.The seriespublishesexpositionsonallaspectsofapplicableandnumericalmathematics,with anemphasisonnewdevelopmentsinthisfast-movingareaofresearch. State-of-the-artmethodsandalgorithmsaswellasmodernmathematicaldescriptions ofphysicalandmechanicalideasarepresentedinamannersuitedtograduateresearch studentsandprofessionalsalike.Soundpedagogicalpresentationisaprerequisite.Itis intendedthatbooksintheserieswillservetoinformanewgenerationofresearchers. Alsointhisseries: 1. APracticalGuidetoPseudospectralMethods,BengtFornberg 2. DynamicalSystemsandNumericalAnalysis,A.M.StuartandA.R.Humphries 3. LevelSetMethodsandFastMarchingMethods,J.A.Sethian 4. TheNumericalSolutionofIntegralEquationsoftheSecondKind,KendallE. Atkinson 5. OrthogonalRationalFunctions,AdhemarBultheel,PabloGonza´lez-Vera,Erik Hendiksen,andOlavNja˚stad 6. TheTheoryofComposites,GraemeW.Milton 7. GeometryandTopologyforMeshGeneration,HerbertEdelsbrunner 8. Schwarz–ChristoffelMapping,TofinA.DriscollandLloydN.Trefethen 9. High-OrderMethodsforIncompressibleFluidFlow,M.O.Deville,P.F.Fischer, andE.H.Mund 10. PracticalExtrapolationMethods,AvramSidi 11. GeneralizedRiemannProblemsinComputationalFluidDynamics,Matania Ben-ArtziandJosephFalcovitz 12. RadialBasisFunctions:TheoryandImplementations,MartinD.Buhmann 13. IterativeKrylovMethodsforLargeLinearSystems,HenkA.vanderVorst 14. SimulatingHamiltonianDynamics,BenLeimkuhlerandSebastianReich 15. CollocationMethodsforVolterraIntegralandRelatedFunctionalEquations, HermannBrunner ii P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 Topology for Computing AFRA J. ZOMORODIAN StanfordUniversity iii    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521836661 © Afra J. Zomorodian 2005 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 —PersistenceofHomology—AfraZomorodian(AfterSalvadorDali) To my parents Ontheleft,adouble-torusanda1-cyclelieonatriangulated2-manifold.Thereisabox-shaped cell-complexabove.Anunknothangsfromthelargebranchofthesaplesswitheringtree.Through someexertion,thetreeidentifiesitselfasamaplebybearingasinglegreenleaf.Adeformedtwo- sphere,atorus,andanonboundingloopformapileinthecenter.Nearthehorizon,a2-manifold isembeddedbyanassociatedheightfield.Itdividesitselfintoregionsusingthe1-cellsofits Morse-Smalecomplex. v P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 vi P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 Contents Preface page xi Acknowledgments xiii 1 Introduction 1 1.1 Spaces 1 1.2 ShapesofSpaces 3 1.3 NewResults 8 1.4 Organization 10 PartOne: Mathematics 2 SpacesandFiltrations 13 2.1 TopologicalSpaces 14 2.2 Manifolds 19 2.3 SimplicialComplexes 23 2.4 AlphaShapes 32 2.5 ManifoldSweeps 37 3 GroupTheory 41 3.1 IntroductiontoGroups 41 3.2 CharacterizingGroups 47 3.3 AdvancedStructures 53 4 Homology 60 4.1 Justification 60 4.2 HomologyGroups 70 4.3 ArbitraryCoefficients 79 5 MorseTheory 83 5.1 TangentSpaces 84 5.2 DerivativesandMorseFunctions 85 5.3 CriticalPoints 86 5.4 StableandUnstableManifolds 88 5.5 Morse-SmaleComplex 90 vii P1:KcS/JZP CY521/Zomorodian-FM CY521/Zomorodian 0521836662 October20,2004 10:47 viii Contents 6 NewResults 94 6.1 Persistence 95 6.2 HierarchicalMorse-SmaleComplexes 105 6.3 LinkingNumber 116 PartTwo: Algorithms 7 ThePersistenceAlgorithms 125 7.1 MarkingAlgorithm 125 7.2 AlgorithmforZ 128 2 7.3 AlgorithmforFields 136 7.4 AlgorithmforPIDs 146 8 TopologicalSimplification 148 8.1 Motivation 148 8.2 ReorderingAlgorithms 150 8.3 Conflicts 153 8.4 TopologyMaps 157 9 TheMorse-SmaleComplexAlgorithm 161 9.1 Motivation 162 9.2 TheQuasiMorse-SmaleComplexAlgorithm 162 9.3 LocalTransformations 166 9.4 Algorithm 169 10 TheLinkingNumberAlgorithm 171 10.1 Motivation 171 10.2 Algorithm 172 PartThree: Applications 11 Software 183 11.1 Methodology 183 11.2 Organization 184 11.3 Development 186 11.4 DataStructures 190 11.5 CView 193 12 Experiments 198 12.1 Three-DimensionalData 198 12.2 AlgorithmforZ 204 2 12.3 AlgorithmforFields 208 12.4 TopologicalSimplification 215 12.5 TheMorse-SmaleComplexAlgorithm 217 12.6 TheLinkingNumberAlgorithm 220 13 Applications 223 13.1 ComputationalStructuralBiology 223 13.2 HierarchicalClustering 227