多面形理论

Preface

ChapterIPreliminaries

I.1Setsandmappings

I.2Partitionsandpermutations

I.3Groupactions

I.4Networks

I.5Notes

ChapterIISurfaces

II.1Polyhedra

II.2Elementaryequivalence

II.3Polyhegons

II.4Orientability

II.5Classification

II.6Notes

ChapterIIIEmbeddlngsofGraphs

III.1Geometricconsideration

III.2Surfaceclosedcurveaxiom

III.3Distinction

III.4Jointtreemodel

III.5Combinatorialproperties

III.6Notes

ChapterIVMathematicalMaps

IV.1Basicpermutations

IV.2Conjugateaxiom

IV.3Transitivity

IV.4Includedangles

IV.5Notes

ChapterVDualityonSurfaces

V.1Dualpartitionofedges

V.2Generaloperation

V.3Basicoperations

V.4Quadrangulations

V.5Notes

ChapterVIInvariantsonBasicClass

VI.1Orientability

VI.2Eulercharacteristic

VI.3Basicequivalence

VI.4Orientablemaps

VI.5Nonorientablemaps

VI.6Notes

ChapterVIIAsymmetrization

VII.1Isomorphisms

VII.2Recognition

VII.3Upperboundofgrouporder

VII.4Determinationofthegroup

VII.5Rootings

VII.6Notes

ChapterVIIIAsymmetrizedCensus

VIII.1Orientableequation

VIII.2Planarmaps

VIII.3Nonorientableequation

VIII.4Grossequation

VIII.5Thenumberofmaps

VIII.6Notes

ChapterIXPetalBundles

IX.1Orientablepetalbundles

IX.2Planarpedalbundles

IX.3Nonorientablepedalbundles

IX.4Thenumberofpedalbundles

IX.5Notes

ChapterXSuperMapsofGenusZero

X.1Plantedtrees

X.2Outerplanargraphs

X.3Hamiltonianplanargraphs

X.4Halingraphs

X.5Notes

ChapterXISymmetricCensus

XI.1Symmetricrelation

XI.2Anapplication

XI.3Symmetricprinciple

XI.4Generalexamples

XI.5Notes

ChapterXIICycleOrientedMaps

XII.1Cycleorientation

XII.2Pan-bouquetsonsurfaces

XII.3Boundarymaps

XII.4Graphsonsurfaces

XII.5Notes

ChapterXIIICensusbyGenus

XIII.1Associatesurfaces

XIII.2Layerdivisionofasurface

XIII.3Handlepolynomials

XIII.4Crosscappolynomials

XIII.5Mapsfromembeddings

XIII.6Graphswithsamepolynomial

XIII.7Notes

ChapterXIVClassicApplications

XIV.1Convexembeddings

XIV.2Rectilinearembeddings

XIV.3Boundarythickness

XIV.4Dehndiagramonknots

XIV.5Pottsmodelsintheoreticalphysics

XIV.6Notes

AppendixIEmbeddingsandMapsofSmallSizeDistributedbyGenus

Ax.I.1Triconnectedcubicgraphs

Ax.I.2Bouquets

Ax.I.3Wheels

Ax.I.4Linkbundles

Ax.I.5Completebipartitegraphs

Ax.I.6Quadregulargraphs

AppendixIIOrientableFormsofSurfacesandTheirNon-orientableGenusPolynomials

Ax.II.1Formsoforientable2β-surfaces

Ax.II.2Nonorientablegenuspolynomials

Bibliography

SubjectIndex

AuthorIndex

　　Thismonographisforaunifiedtheoryofsurfaces,embeddingsandmapsallconsideredaspolyhedraviathejointtreemodalwhichwasinitiatedfromtheauthorsarticlesintheseventiesoflastcenturyandhasbeenbasicallydevelopedinrecentdecades.Completeinvariantsforeachclassificationaretopologically,combinatoriallyorisomorphicallyextracted.Anumberofcountingpolynomialsincludinghandleandcrosscappolynomialsarepresented.Inparticular,anappendixservesastheexhaustivecountingsupermaps(rootedandnonrooted)includingthesepolynomialswithundergraphsofsmallsizeforthereadersdigests.Althoughthebookismainlyforresearchersinmathematics,theoreticalphysics,chemistry,biologyandsomeothersrelated,thebasicpartineachchaptercanalsobechosenforgraduatesandcollegeteachersasreferences.