Knots

572 Pages·2003·2.725 MB·English

by  Gerhard Burde, Heiner Zieschang

de Gruyter Studies in Mathematics 5
Editors: Carlos Kenig · Andrew Ranicki · Michael Röckner
de Gruyter Studies in Mathematics
1 Riemannian Geometry, 2nd rev. ed., Wilhelm P.A.Klingenberg
2 Semimartingales, Michel Me´tivier
3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup
4 Spaces of Measures, Corneliu Constantinescu
5 Knots, Gerhard Burde and Heiner Zieschang
6 Ergodic Theorems, Ulrich Krengel
7 Mathematical Theory of Statistics, Helmut Strasser
8 Transformation Groups, Tammo tom Dieck
9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii
10 Analyticity in Infinite Dimensional Spaces, Michel Herve´
11 Elementary Geometry in Hyperbolic Space, Werner Fenchel
12 Transcendental Numbers, Andrei B. Shidlovskii
13 Ordinary Differential Equations, Herbert Amann
14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and
Francis Hirsch
15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine
16 Rational Iteration, Norbert Steinmetz
17 Korovkin-typeApproximationTheoryanditsApplications,FrancescoAltomare
and Michele Campiti
18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev
19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima,
Yoichi Oshima and Masayoshi Takeda
20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R.Bloom
and Herbert Heyer
21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov
22 Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii,
Victor E. Shatalov and Boris Yu. Sternin
23 Probability Theory, Heinz Bauer
24 Variational Methods for Potential Operator Equations, Jan Chabrowski
25 The Structure of Compact Groups, Karl H. Hofmann and Sidney A. Morris
26 Measure and Integration Theory, Heinz Bauer
27 Stochastic Finance, Hans Föllmer and Alexander Schied
28 Painleve´ Differential Equations in the Complex Plane, Valerii I. Gromak, Ilpo
Laine and Shun Shimomura
29 Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel
and Jakob Nielsen
Gerhard Burde · Heiner Zieschang
Knots
Second Revised and Extended Edition
≥
Walter de Gruyter
Berlin · New York 2003
Authors
GerhardBurde HeinerZieschang
FachbereichMathematik(Fach187) FakultätfürMathematik
UniversitätFrankfurtamMain Ruhr-UniversitätBochum
Robert-Mayer-Str.6(cid:1)10 Universitätsstr.150
60325Frankfurt/Main 44801Bochum
Germany Germany
SeriesEditors
CarlosE.Kenig AndrewRanicki MichaelRöckner
DepartmentofMathematics DepartmentofMathematics FakultätfürMathematik
UniversityofChicago UniversityofEdinburgh UniversitätBielefeld
5734UniversityAve MayfieldRoad Universitätsstraße25
Chicago,IL60637,USA EdinburghEH93JZ,Scotland 33615Bielefeld,Germany
MathematicsSubjectClassification2000:57-02;57M25,20F34,20F36
Keywords:knots;links;fibredknots;torusknots;factorization;braids;branchedcoverings;Montesinos
links; knot groups; Seifert surfaces; Alexander polynomials; Seifert matrices; cyclic periods of knots;
homflypolynomial
With184figures
(cid:1)(cid:1) Printedonacid-freepaperwhichfallswithintheguidelinesoftheANSI
toensurepermanenceanddurability.
LibraryofCongressCataloging-in-PublicationData
Burde,Gerhard,1931(cid:1)
Knots/GerhardBurde,HeinerZieschang.(cid:1)2ndrev.andextended
ed.
p. cm.(cid:1)(DeGruyterstudiesinmathematics;5)
Includesbibliographicalreferencesandindex.
ISBN3110170051(cloth:alk.paper)
I.Zieschang,Heiner. II.Title. III.Series.
QA612.2.B87 2002
514(cid:2).224(cid:1)dc21 2002034764
ISBN3110170051
BibliographicinformationpublishedbyDieDeutscheBibliothek
DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataisavailableintheInternetat(cid:3)http://dnb.ddb.de(cid:4).
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Preface to the First Edition
The phenomenon of a knot is a fundamental experience in our perception of three
dimensionalspace. Whatisspecialaboutknotsisthattheyrepresentatrulyintrinsic
andessentialqualityof3-spaceaccessibletointuitiveunderstanding. Noarbitrariness
likethechoiceofametricmarsthenatureofaknot–atrefoilknotwillbeuniversally
recognizable wherever the basic geometric conditions of our world exist. (One is
temptedtoproposeitasanemblemofouruniverse.)
There is no doubt that knots hold an important – if not crucial – position in the
theory of 3-dimensional manifolds. As a subject for a mathematical textbook they
serve a double purpose. They are excellent introductory material to geometric and
algebraictopology,helpingtounderstandproblemsandtorecognizeobstructionsin
thisfield. Ontheotherhandtheypresentthemselvesasreadyandcopioustestmaterial
fortheapplicationofvariousconceptsandtheoremsintopology.
Thefirstninechapters(exceptingthesixth)treatstandardmaterialofclassicalknot
theory. Theremainingchaptersaredevotedtomoreorlessspecialtopicsdepending
ontheinterestandtasteoftheauthorsandwhattheybelievedtobeessentialandalive.
Thesubjectsmight,ofcourse,havebeenselectedquitedifferentlyfromtheabundant
wealthofpublicationsinknottheoryduringthelastdecades.
Wehavestuckthroughoutthisbookmainlytotraditionaltopicsofclassicalknot
theory. Linkshavebeenincludedwheretheycomeinnaturally. Higher-dimensional
knottheoryhasbeencompletelyleftout–evenwhereithasabearingon3-dimensional
knotssuchassliceknots. Thethemeofsurgeryhasbeenratherneglected–excepting
Chapter15. WildknotsandAlgebraicknotsaremerelymentioned.
Thisbookmaybereadbystudentswithabasicknowledgeinalgebraictopology
– at least the first four chapters will present no serious difficulties to them. As the
book proceeds certain fundamental results on 3-manifolds are used – such as the
Papakyriakopoulostheorems. ThetheoremsarestatedinAppendixBandreferences
aregivenwhereproofsmaybefound. Thereseemedtobenopointinaddinganother
presentation of these things. The reader who is not familiar with these theorems is,
however, well advised to interrupt the reading to study them. At some places the
theoryofsurfacesisneeded–severalresultsofNielsenareapplied. Proofsofthese
may be read in [ZVC 1980], but taking them for granted will not seriously impair
theunderstandingofthisbook. Wheneverpossiblewehavegivencompleteandself-
contained proofs at the most elementary level possible. To do this we occasionally
refrained from applying a general theorem but gave a simpler proof for the special
caseinhand.
Thereare,ofcourse,manypertinentandinterestingfactsinknottheory–especially
initsrecentdevelopment–thatweredefinitelybeyondthescopeofsuchatextbook.
Tobecomplete–eveninaspecialfieldsuchasknots–isimpossibletodayandwas
notaimedat. Wetriedtokeepupwithimportantcontributionsinourbibliography.
vi PrefacetotheFirstEdition
Therearenotmanytextbooksonknots. Reidemeister’s“Knotentheorie”wascon-
ceivedforadifferentpurposeandlevel;Neuwirth’sbook“KnotGroups”andHillman’s
monograph“AlexanderIdealsofLinks”haveamorespecialisedandalgebraicinterest
in mind. In writing this book we had, however, to take into consideration Rolfsen’s
remarkablebook“KnotsandLinks”. Wetriedtoavoidoverlappingsinthecontents
andthemannerofpresentation. Inparticular,wethoughtitfutiletoproduceanother
setofdrawingsofknotsandlinksuptotencrossings–orevenmore. Theycan–in
perfectbeauty–beviewedinRolfsen’sbook. Knotswithlessthantencrossingshave
beenaddedinAppendixDasaminimumofreadyillustrativematerial. Thetablesof
knotinvariantshavealsobeendevisedinawaywhichoffersatleastsomethingnew.
Figuresareplentifulbecausewethinkthemnecessaryandhopethemtobehelpful.
FinallywewishtoexpressourgratitudetoColinMaclachlanwhoreadthemanu-
scriptandexpurgateditfromthegrosserlapsuslinguae(thissentencewascomposed
without his supervision). We are indebted to U. Lüdicke and G.Wenzel who wrote
thecomputerprogramsandcarriedoutthecomputationsofamajorpartoftheknot
invariants listed in the tables. We are grateful to U. Dederek-Breuer who wrote the
programforfilingandsortingthebibliography. WealsowanttothankMrs.A.Huck
and Mrs. M. Schwarz for patiently typing, re-typing, correcting and re-correcting
abominablemanuscripts.
Frankfurt(Main)/Bochum,Summer1985 GerhardBurde
HeinerZieschang
Preface to the Second Edition
The text has been revised, some mistakes have been eliminated and Chapter 15 has
beenbroughtuptodate,especiallytakingintoaccounttheGordon–LueckeTheorem
onknotcomplements,althoughwehavenotincludedaproof. Chapter16wasadded,
presentinganintroductiontotheHOMFLYpolynomial,andincludingaself-contained
accountofthefundamentalfactsaboutHeckealgebras. AproofofMarkov’stheorem
wasaddedinChapter10onbraids. Wealsotriedtobringthebibliographyuptodate.
In view of the vast amount of recent and pertinent contributions even approximate
completenesswasoutofthequestion.
We have decided not to deal withVassiliev invariants, quantum group invariants
andhyperbolicstructuresonknotcomplements, sinceathoroughtreatmentofthese
topicswouldgofarbeyondthespaceatourdisposal. Adequateintroductorysurveys
onthesetopicsareavailableelsewhere.
Sincethefirsteditionofthisbookin1985,aseriesofbooksonknotsandrelated
topics have appeared. We mention especially: [Kauffman 1987, 1991], [Kawauchi
1996],[Murasugi1996],[Turaev1994],[Vassiliev1999].
Our heartfelt thanks go to Marlene Schwarz and Jörg Stümke for producing the
LATEX-fileandtoRichardWeidmannforproof-reading. Wealsothanktheeditorsfor
theirpatienceandpleasantcooperation,andIreneZimmermannforhercarefulwork
onthefinallayout.
Frankfurt(Main)/Bochum,2002 GerhardBurde
HeinerZieschang

Contents
1 KnotsandIsotopies 1
A Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
B EquivalenceofKnots . . . . . . . . . . . . . . . . . . . . . . . . . . 4
C KnotProjections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
D GlobalGeometricProperties . . . . . . . . . . . . . . . . . . . . . . 11
E HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
F Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 GeometricConcepts 15
A GeometricPropertiesofProjections . . . . . . . . . . . . . . . . . . 15
B SeifertSurfacesandGenus . . . . . . . . . . . . . . . . . . . . . . . 17
C CompanionKnotsandProductKnots . . . . . . . . . . . . . . . . . 19
D Braids,Bridges,Plats . . . . . . . . . . . . . . . . . . . . . . . . . . 22
E SliceKnotsandAlgebraicKnots . . . . . . . . . . . . . . . . . . . . 25
F HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
G Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 KnotGroups 30
A Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
B WirtingerPresentation . . . . . . . . . . . . . . . . . . . . . . . . . 32
C PeripheralSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
D KnotsonHandlebodies . . . . . . . . . . . . . . . . . . . . . . . . . 42
E TorusKnots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
F AsphericityoftheKnotComplement . . . . . . . . . . . . . . . . . . 48
G HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
H Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 CommutatorSubgroupofaKnotGroup 52
A ConstructionofCyclicCoverings. . . . . . . . . . . . . . . . . . . . 52
B StructureoftheCommutatorSubgroup. . . . . . . . . . . . . . . . . 55
C ALemmaofBrownandCrowell . . . . . . . . . . . . . . . . . . . . 57
D ExamplesandApplications . . . . . . . . . . . . . . . . . . . . . . . 59
E CommutatorSubgroupsofSatellites . . . . . . . . . . . . . . . . . . 62
F HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
G Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
x Contents
5 FibredKnots 68
A FibrationTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B FibredKnots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C ApplicationsandExamples . . . . . . . . . . . . . . . . . . . . . . . 73
D HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
E Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 ACharacterizationofTorusKnots 79
A ResultsandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B ProofoftheMainTheorem . . . . . . . . . . . . . . . . . . . . . . . 81
C RemarksontheProof . . . . . . . . . . . . . . . . . . . . . . . . . . 87
D HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
E Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 FactorizationofKnots 91
A CompositionofKnots . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B UniquenessoftheDecompositionintoPrimeKnots: Proof . . . . . . 96
C FibredKnotsandDecompositions . . . . . . . . . . . . . . . . . . . 99
D HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
E Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8 CyclicCoveringsandAlexanderInvariants 103
A TheAlexanderModule . . . . . . . . . . . . . . . . . . . . . . . . . 103
B InfiniteCyclicCoveringsandAlexanderModules . . . . . . . . . . . 104
C HomologicalPropertiesofC∞ . . . . . . . . . . . . . . . . . . . . . 109
D AlexanderPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 111
E FiniteCyclicCoverings . . . . . . . . . . . . . . . . . . . . . . . . . 117
F HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
G Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9 FreeDifferentialCalculusandAlexanderMatrices 125
A RegularCoveringsandHomotopyChains . . . . . . . . . . . . . . . 125
B FoxDifferentialCalculus . . . . . . . . . . . . . . . . . . . . . . . . 127
C CalculationofAlexanderPolynomials . . . . . . . . . . . . . . . . . 129
D AlexanderPolynomialsofLinks . . . . . . . . . . . . . . . . . . . . 134
E FiniteCyclicCoveringsAgain . . . . . . . . . . . . . . . . . . . . . 137
F HistoryandSources . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
G Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10 Braids 142
A TheClassificationofBraids. . . . . . . . . . . . . . . . . . . . . . . 142
B NormalFormandGroupStructure . . . . . . . . . . . . . . . . . . . 150
C ConfigurationSpacesandBraidGroups . . . . . . . . . . . . . . . . 155
D BraidsandLinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159