An Introduction to Knot Theory: 175 (Graduate Texts in Mathematics, 175) - Hardcover

Lickorish, W.B.Raymond

Hardcover

ISBN 10: 038798254X ISBN 13: 9780387982540

Publisher: Springer, 1997

Synopsis

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.

About this title

From the Back Cover

This volume is an introduction to mathematical knot theory - the theory of knots and links of simple closed curves in three-dimensional space. It consists of a selection of topics that graduate students have found to be a successful introduction to the field. Three distinct techniques are employed: geometric topology manoeuvres; combinatorics; and algebraic topology. Each topic is developed until significant results are achieved, and chapters end with exercises and brief accounts of state-of-the-art research. What may reasonably be referred to as knot theory has expanded enormously over the last decade, and while the author describes important discoveries from throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds - as well as generalisations and applications of the Jones polynomial - are also included, presented in an easily understandable style. Thus, this constitutes a comprehensive introduction to the field, presenting modern developments in the context of classical material. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are plentiful and well done. Written by an internationally known expert in the field, this volume will appeal to graduate students, mathematicians, and physicists with a mathematical background who wish to gain new insights in this area.

Edition details

• Publisher: Springer
• Publication date: 1997
• Language: English
• ISBN 10: 038798254X
• ISBN 13: 9780387982540
• Binding: Hardcover
• Number of pages: 214