Knots

This book is an introduction to classical knot theory. Topics covered include: different constructions of knots, knot diagrams, knot groups, fibred knots, characterisation of torus knots, prime decomposition of knots, cyclic coverings and Alexander polynomials and modules together with the free differential calculus, braids, branched coverings and knots, Montesinos links, representations of knot groups, surgery of 3-manifolds and knots.

Knot theory has expanded enormously since the first edition of this book published in 1985. A special feature of this second completely revised and extended edition is the introduction to two new constructions of knot invariants, namely the Jones and homfly polynomials and the Vassiliev invariants.

The book contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory.

Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups and some basic results of combinatorial group theory are assumed to be known. The text is accessible to advanced undergraduate and graduate students in mathematics.

Résumé

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community.

Please submit any book proposals to Niels Jacob.

Contenu

• Knots and Isotopies
• Geometric Concepts
• Knot Groups
• Commutator Subgroup of a Knot Group
• Fibred Knots
• A Characterization of Torus Knots
• Factorization of Knots
• Cyclic Coverings and Alexander Invariants
• Free Differential Calculus and Alexander Matrices
• Braids
• Manifolds as Branched Coverings
• Montesinos Links
• Quadratic Forms of a Knot
• Representations of Knot Groups
• Knots, Knot Manifolds, and Knot Groups
• The 2-variable skein polynomial Appendix A: Algebraic Theorems
• Appendix B: Theorems of 3-dimensional Topology
• Appendix C: Tables
• Appendix D: Knot Projections
• List of Authors According to Codes
• List of Papers According to Codes
• Glossary of Symbols
• Author Index
• Subject Index