Introduction to Knot Theory.

Abstract

Prerequisites.- I * Knots and Knot Types.- 1. Definition of a knot.- 2. Tame versus wild knots.- 3. Knot projections.- 4. Isotopy type, amphicheiral and invertible knots.- II * The Fundamental Group.- 1. Paths and loops.- 2. Classes of paths and loops.- 3. Change of basepoint.- 4. Induced homomorphisms of fundamental groups.- 5. Fundamental group of the circle.- III * The Free Groups.- 1. The free group F[A].- 2. Reduced words.- 3. Free groups.- IV * Presentation of Groups.- 1. Development of the presentation concept.- 2. Presentations and presentation types.- 3. The Tietze theorem.- 4. Word subgroups and the associated homomorphisms.- 5. Free abelian groups.- V * Calculation of Fundamental Groups.- 1. Retractions and deformations.- 2. Homotopy type.- 3. The van Kampen theorem.- VI * Presentation of a Knot Group.- 1. The over and under presentations.- 2. The over and under presentations, continued.- 3. The Wirtinger presentation.- 4. Examples of presentations.- 5. Existence of nontrivial knot types.- VII * The Free Calculus and the Elementary Ideals.- 1. The group ring.- 2. The free calculus.- 3. The Alexander matrix.- 4. The elementary ideals.- VIII * The Knot Polynomials.- 1. The abelianized knot group.- 2. The group ring of an infinite cyclic group.- 3. The knot polynomials.- 4. Knot types and knot polynomials.- IX * Characteristic Properties of the Knot Polynomials.- 1. Operation of the trivialize.- 2. Conjugation.- 3. Dual presentations.- Appendix I. Differentiable Knots are Tame.- Appendix II. Categories and groupoids.- Appendix III. Proof of the van Kampen theorem.- Guide to the Literature.