Introduction to Knot Theory

1. Front Matter

   Pages i-x

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2. Prerequisites

   • Richard H. Crowell, Ralph H. Fox

   Pages 1-2

3. Knots and Knot Types

   • Richard H. Crowell, Ralph H. Fox

   Pages 3-12

4. The Fundamental Group

   • Richard H. Crowell, Ralph H. Fox

   Pages 13-30

5. The Free Groups

   • Richard H. Crowell, Ralph H. Fox

   Pages 31-36

6. Presentation of Groups

   • Richard H. Crowell, Ralph H. Fox

   Pages 37-51

7. Calculation of Fundamental Groups

   • Richard H. Crowell, Ralph H. Fox

   Pages 52-71

8. Presentation of a Knot Group

   • Richard H. Crowell, Ralph H. Fox

   Pages 72-93

9. The Free Calculus and the Elementary Ideals

   • Richard H. Crowell, Ralph H. Fox

   Pages 94-109

10. The Knot Polynomials

    • Richard H. Crowell, Ralph H. Fox

    Pages 110-133

11. Characteristic Properties of the Knot Polynomials

    • Richard H. Crowell, Ralph H. Fox

    Pages 134-145

12. Back Matter

    Pages 147-182

    PDF

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 differential geometry, to name some of the more prominent ones. It had its origins in the mathematical theory of electricity and in primitive atomic physics, and there are hints today of new applications in certain branches of chemistryJ The outlines of the modern topological theory were worked out by Dehn, Alexander, Reidemeister, and Seifert almost thirty years ago. As a subfield of topology, knot theory forms the core of a wide range of problems dealing with the position of one manifold imbedded within another. This book, which is an elaboration of a series of lectures given by Fox at Haverford College while a Philips Visitor there in the spring of 1956, is an attempt to make the subject accessible to everyone. Primarily it is a text­ book for a course at the junior-senior level, but we believe that it can be used with profit also by graduate students. Because the algebra required is not the familiar commutative algebra, a disproportionate amount of the book is given over to necessary algebraic preliminaries.

• Knot theory
• Manifold
• Topology
• algebra
• geometry
• mathematics

• Department of Mathematics, Dartmouth College, Hanover, USA

  Richard H. Crowell

• Princeton University, Princeton, USA

  Ralph H. Fox

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