W. B. Raymond Lickorish (auth.)9780387982540, 038798254X
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 :
Front Matter….Pages i-x
A Beginning for Knot Theory….Pages 1-14
Seifert Surfaces and Knot Factorisation….Pages 15-22
The Jones Polynomial….Pages 23-31
Geometry of Alternating Links….Pages 32-40
The Jones Polynomial of an Alternating Link….Pages 41-48
The Alexander Polynomial….Pages 49-65
Covering Spaces….Pages 66-78
The Conway Polynomial, Signatures and Slice Knots….Pages 79-92
Cyclic Branched Covers and the Goeritz Matrix….Pages 93-102
The Arf Invariant and the Jones Polynomial….Pages 103-109
The Fundamental Group….Pages 110-122
Obtaining 3-Manifolds by Surgery on S 3 ….Pages 123-132
3-Manifold Invariants from the Jones Polynomial….Pages 133-145
Methods for Calculating Quantum Invariants….Pages 146-165
Generalisations of the Jones Polynomial….Pages 166-178
Exploring the HOMFLY and Kauffman Polynomials….Pages 179-192
Back Matter….Pages 193-204 |
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