Book contents
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 A Topologist's Toolkit
- 3 Link Diagrams
- 4 Constructions and Decompositions of Links
- 5 Spanning Surfaces and Genus
- 6 Matrix Invariants
- 7 The Alexander–Conway Polynomial
- 8 Rational Tangles
- 9 More Polynomials
- 10 Closed Braids and Arc Presentations
- Appendix A Knot Diagrams
- Appendix B Numerical Invariants
- Appendix C Properties
- Appendix D Polynomials
- Appendix E Polygon Coordinates
- Appendix F Family Properties
- Bibliography
- Index
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 A Topologist's Toolkit
- 3 Link Diagrams
- 4 Constructions and Decompositions of Links
- 5 Spanning Surfaces and Genus
- 6 Matrix Invariants
- 7 The Alexander–Conway Polynomial
- 8 Rational Tangles
- 9 More Polynomials
- 10 Closed Braids and Arc Presentations
- Appendix A Knot Diagrams
- Appendix B Numerical Invariants
- Appendix C Properties
- Appendix D Polynomials
- Appendix E Polygon Coordinates
- Appendix F Family Properties
- Bibliography
- Index
Summary
Knot theory is the study of embeddings of circles in space. It is a subject in which naturally occurring questions are often so simple to state that they can be explained to a child, yet finding answers may require ideas from the forefront of research. It is a subject of both depth and subtlety.
The subject started to develop systematically in the late nineteenth century, and has seen explosive growth during the last 20 years. Even so, many of the ideas and recent results can be explained without the use of advanced mathematical technology and can be made accessible to undergraduates.
The study of knots and links is part of topology, the branch of geometry that deals with flexible and deformable spaces. A topologist concentrates on properties that remain unchanged by continuous transformations that can be undone. In a first topology course students meet:
topological spaces and their main properties – open and closed sets, continuity, homeomorphisms, connectedness, compactness;
topological structures – concrete topology, discrete topology, metric spaces;
topological constructions – product spaces, quotient spaces;
topological invariants like the Euler characteristic;
examples such as the classification of surfaces.
These are all intrinsic properties of a space. Another aspect of topology is the study of the ways that spaces can sit inside one another – the embedding problem. Knot theory is a perfect introduction to this and makes a good second course in topology.
This book gives an introductory account of knot theory from the geometric viewpoint. It provides a good grounding in the subject for anyone wishing to do research in knot theory or geometric topology, but also stands on its own as a course suitable for final year undergraduates.
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- Information
- Knots and Links , pp. xi - xivPublisher: Cambridge University PressPrint publication year: 2004