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Introduction to Knot Theory

Knot theory is a kind of geometry, and one whose appeal is very direct because the objects studied are perceivable and tangible in everyday physical space. It is a meeting ground of such diverse branches of mathematics as group theory, matrix theory, number theory, algebraic geometry, and differenti...

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Bibliographic Details
Main Authors: Crowell, R. H., Fox, R. H. (Author)
Format: eBook
Language:English
Published: New York, NY Springer New York 1963, 1963
Edition:1st ed. 1963
Series:Graduate Texts in Mathematics
Subjects:
Topology
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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

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