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Introduction to Knot Theory
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Catalogue Information
Field name
Details
Dewey Class
514
Title
Introduction to Knot Theory ([EBook] /) / by Richard H. Crowell, Ralph H. Fox.
Author
Crowell, Richard H.
Added Personal Name
Fox, Ralph H.
author.
Other name(s)
SpringerLink (Online service)
Publication
New York, NY : : Springer New York, , 1963.
Physical Details
X, 182 p. : online resource.
Series
Graduate texts in mathematics
0072-5285 ; ; 57
ISBN
9781461299356
Summary Note
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 note
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.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-1-4612-9935-6
Links to Related Works
Subject References:
Mathematics
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Topology
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Authors:
Crowell, Richard H.
.
Fox, Ralph H.
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Corporate Authors:
SpringerLink (Online service)
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Series:
Graduate texts in mathematics
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GTM
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Classification:
514
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