Topology of Surfaces, Knots, and Manifolds

Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds. A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics.

Stephan C. Carlson is the author of Topology of Surfaces, Knots, and Manifolds, published by Wiley.

Introduction and Intuitive Ideas

1

(25)

What Is Topology?

1

(6)

Getting More Formal

7

(4)

Functions and Homeomorphisms

11

(7)

Constructing Objects from Plane Models

18

(4)

Chapter Summary

22

(1)

Supplementary Exercises

22

(3)

Manifolds

25

(20)

Basic Definitions and Curves

25

(7)

Surfaces

32

(2)

Compact Surfaces and Plane Models

34

(6)

Three-dimensional Manifolds

40

(3)

Chapter Summary

43

(1)

Supplementary Exercises

44

(1)

Classification of Compact Surfaces

45

(23)

Connected Sums: New Surfaces from Old

45

(3)

The Algebra of Surfaces

48

(12)

The Classification Theorem: Preliminary Ideas

60

(3)

The Proof of the Classification Theorem

63

(3)

Chapter Summary

66

(1)

Supplementary Exercises

67

(1)

Putting More Structure on the Surfaces

68

(31)

Tilings on Surfaces and the Euler Characteristic

68

(11)

Patterns, Complexes, and Regularity

79

(11)

Coloring Maps on Surfaces

90

(7)

Chapter Summary

97

(1)

Supplementary Exercises

97

(2)

Graphs and Topology

99

(1)

What is a Graph?

99

(22)

Paths in Graphs

104

(6)

Embedding Graphs in Surfaces

110

(8)

Chapter Summary

118

(1)

Supplementary Exercises

119

(2)

Knot Theory

121

(30)

Knots and Links: the Basics

122

(7)

Oriented Links and Linking Number

129

(4)

Tricolorability of Knots and Links

133

(4)

Polynomial Invariants of Knots and Links

137

(10)

Chapter Summary

147

(1)

Supplementary Exercises

148

(3)

Appendix

151

(1)

References and Suggestions for Further Study

151

(4)

Index

155

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Topology of Surfaces, Knots, and Manifolds

Summary

Author Biography

Table of Contents

Supplemental Materials