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CAMBRIDGEMONOGRAPHSON
APPLIEDANDCOMPUTATIONAL
MATHEMATICS
SeriesEditors
P.G.CIARLET,A.ISERLES,R.V.KOHN,M.H.WRIGHT
16 Topology for Computing
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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
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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.
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—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.
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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
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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
