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Prerequisites 1

Chapter Ⅰ Knots and Knot Types 3

1.Definition of a knot 3

2.Tame versus wild knots 5

3.Knot projections 6

4.Isotopy type,amphicheiral and invertible knots 8

Chapter Ⅱ The Fundamental Group 13

Introduction 13

1.Paths and loops 14

2.Classes of paths and loops 15

3.Change of basepoint 21

4.Induced homomorphisms of fundamental groups 22

5.Fundamental group of the circle 24

Chapter Ⅲ The Free Groups 31

Introduction 31

1.The free group F[？] 31

2.Reduced words 32

3.Free groups 35

Chapter Ⅳ Presentation of Groups 37

Introduction 37

1.Development of the presentation concept 37

2.Presentations and preeentation types 39

3.The Tietze theorem 43

4.Word subgroups and the associated homomorphisms 47

5.Free abelian groups 50

Chapter Ⅴ Calculation of Fundamental Groups 52

Introduction 52

1.Retractions and deformations 54

2.Homotopy type 62

3.The van Kampen theorem 63

Chapter Ⅵ Presentation of a Knot Group 72

Introduction 72

1.The over and under presentations 72

2.The over and under presentations,continued 78

3.The Wirtinger presentation 86

4.Examples of presentations 87

5.Existence of nontrivial knot types 90

Chapter Ⅶ The Free Calculus and the Elementary Ideals 94

Introduction 94

1.The group ring 94

2.The free calculus 96

3.The Alexander matrix 100

4.The elementary ideals 101

Chapter Ⅷ The Knot Polynomials 110

Introduction 110

1.The abelianized knot group 111

2.The group ring of an infinite cyclic group 113

3.The knot polynomials 119

4.Knot types and knot polynomials 123

Chapter Ⅸ Characteristic Properties of the Knot Polynomials 134

Introduction 134

1.Operation of the trivializer 134

2.Conjugation 136

3.Dual presentations 137

Appendix Ⅰ.Differentiable Knots are Tame 147

Appendix Ⅱ.Categories and groupeids 153

Appendix Ⅲ.Proof of the van Kampen theorem 156

Guide to the Literature 161

Bibliography 165

Index 178