Knot Theory, Volume 24

Front Cover
Cambridge University Press, 1993 - Mathematics - 240 pages
Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.
 

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Contents

A CENTURY OF KNOT THEORY
1
What Is A Knot?
11
Diagrams and Projections
24
Colorings
32
Matrices Labelings
42
The Alexander Polynomial
48
GEOMETRIC TECHNIQUES
55
Surfaces and Homeomorphisms
62
The Fundamental Group
105
Seifert Matrices and the Alexander
114
Knot Groups and the Alexander
123
NUMERICAL INVARIANTS
129
Braids and Bridges
135
Relations between the Numerical
141
SYMMETRIES OF KNOTS 151 Chapter 8 SYMMETRIES OF KNOTS
152
The Murasugi Conditions
167

Surgery on Surfaces
73
ALGEBRAIC TECHNIQUES
83
Knots and Groups
89
Equations in Groups and the Group
99
Applications of the Murasugi
173
HIGHDIMENSIONAL KNOT
179
Threedimensional Crosssections
186
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