Introduction to Knot Theory

Contents......Page 7
Prerequisites......Page 9
1. Definition of a knot......Page 11
2. Tame versus wild knots......Page 13
3. Knot projections......Page 14
4. Isotopy type, amphicheiral and invertible knots......Page 16
Introduction......Page 21
1. Paths and loops......Page 22
2. Classes of paths and loops......Page 23
3. Change of basepoint......Page 29
4. Induced homomorphisms of fundamental groups......Page 30
5. Fundamental group of the circle......Page 32
1. The free group F[A]......Page 39
2. Reduced words......Page 40
3. Free groups......Page 43
1. Development of the presentation concept......Page 45
2. Presentations and presentation types......Page 47
3. The Tietze theorem......Page 51
4. Word subgroups and the associated homomorphisms......Page 55
5. Free abelian groups......Page 58
Introduction......Page 60
1. Retractions and deformations......Page 62
2. Homotopv type......Page 70
3. The van Kampen theorem......Page 71
1. The over and under presentations......Page 80
2. The over and under presentations, continued......Page 86
3. The Wirtinger presentation......Page 94
4. Examples of presentations......Page 95
5. Existence of nontrivial knot types......Page 98
1. The group ring......Page 102
2. The free calculus......Page 104
3. The Alexander matrix......Page 108
4. The elementary ideals......Page 109
Introduction......Page 118
1. The abelianized knot group......Page 119
2. The group ring of an infinite cyclic group......Page 121
3. The knot polynomials......Page 127
4. Knot types and knot polynomials......Page 131
1. Operation of the trivializer......Page 142
2. Conjugation......Page 144
3. Dual presentations......Page 145
Appendix I. Differentiable Knots are Tame......Page 154
Appendix II. Categories and groupoids......Page 160
Appendix III. Proof of the van Kampen theorem......Page 163
Guide to the Literature......Page 168
Bibliography......Page 172
Index......Page 185