《INTRODUCTION TO KNOT THEORY》

作者 RICHARD H. CROWELL RALPH H. FO 编者
出版 SPRINGER-VERLAG NEW YORK HEIDELBERG BERLIN
参考页数 182
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INTRODUCTION TO KNOT THEORY PDF电子版封面 RICHARD H. CROWELL RALPH H. FO

Prerequisites1

Chapter ⅠKnots and Knot Types3

1.Definition of a knot3

2.Tame versus wild knots5

3.Knot projections6

4.Isotopy type,amphicheiral and invertible knots8

Chapter ⅡThe Fundamental Group13

Introduction13

1.Paths and loops14

2.Classes of paths and loops15

3.Change of basepoint21

4.Induced homomorphisms of fundamental groups22

5.Fundamental group of the circle24

Chapter ⅢThe Free Groups31

Introduction31

1.The free group F[？]31

2.Reduced words32

3.Free groups35

Chapter ⅣPresentation of Groups37

Introduction37

1.Development of the presentation concept37

2.Presentations and preeentation types39

3.The Tietze theorem43

4.Word subgroups and the associated homomorphisms47

5.Free abelian groups50

Chapter ⅤCalculation of Fundamental Groups52

Introduction52

1.Retractions and deformations54

2.Homotopy type62

3.The van Kampen theorem63

Chapter ⅥPresentation of a Knot Group72

Introduction72

1.The over and under presentations72

2.The over and under presentations,continued78

3.The Wirtinger presentation86

4.Examples of presentations87

5.Existence of nontrivial knot types90

Chapter ⅦThe Free Calculus and the Elementary Ideals94

Introduction94

1.The group ring94

2.The free calculus96

3.The Alexander matrix100

4.The elementary ideals101

Chapter ⅧThe Knot Polynomials110

Introduction110

1.The abelianized knot group111

2.The group ring of an infinite cyclic group113

3.The knot polynomials119

4.Knot types and knot polynomials123

Chapter ⅨCharacteristic Properties of the Knot Polynomials134

Introduction134

1.Operation of the trivializer134

2.Conjugation136

3.Dual presentations137

Appendix Ⅰ.Differentiable Knots are Tame147

Appendix Ⅱ.Categories and groupeids153

Appendix Ⅲ.Proof of the van Kampen theorem156

Guide to the Literature161

Bibliography165

Index178

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