This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. The study of knots can be given some motivation in terms of applications in molecular biology or by reference to paral­ lels in equilibrium statistical mechanics or quantum field theory. Here, however, knot theory is considered as part of geometric topology. Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge­ ometry of three-dimensional space can be explored by knotting phenomena using precise mathematics. The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology. The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding. In particular, a knowledge of the very basic ideas of the fundamental group and of a simple homology theory is assumed; it is, after all, more important to know about those topics than about the intricacies of knot theory.

Table of Contents

Chapter 1. A Beginning for Knot Theory

Chapter 2. Seifert Surfaces and Knot Factorisation

Chapter 3. The Jones Polynomial

Chapter 4. Geometry of Alternating Links

Chapter 5. The Jones Polynomial of an Alternating Link

Chapter 6. The Alexander Polynomial

Chapter 7. Covering Spaces

Chapter 8. The Conway Polynomial, Signatures and Slice Knots

Chapter 9. Cyclic Branched Covers and the Goeritz Matrix

Chapter 10. The Arf Invariant and the Jones Polynomial

Chapter 11. The Fundamental Group

Chapter 12. Obtaining 3-Manifolds by Surgery on S3

Chapter 13. 3-Manifold Invariants From The Jones Polynomial

Chapter 14. Methods for Calculating Quantum Invariants

Chapter 15. Generalisations of the Jones Polynomial

Chapter 16. Exploring the HOMFLY and Kauffman Polynomials

About the Author

William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow of Pembroke College, Cambridge. His research interests include topology and knot theory.