Introduction to Knot Theory / by R. H. Crowell, R. H. Fox.

Format:

Book

Author/Creator:

Crowell, Richard H., author.

Fox, Ralph H. (Ralph Hartzler), 1913-1973, author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 2197-5612 ; 57.

Graduate Texts in Mathematics, 2197-5612 ; 57

Language:

English

Subjects (All):

Topology.

Local Subjects:

Topology.

Physical Description:

1 online resource (X, 182 pages).

Edition:

First edition 1963.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1963.

System Details:

text file PDF

Summary:

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.

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.

Other Format:

Printed edition:

ISBN:

9781461299356

Access Restriction:

Restricted for use by site license.