Knots / Edition 2

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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.
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Product Details

  • ISBN-13: 9783110170054
  • Publisher: De Gruyter
  • Publication date: 2/28/2003
  • Series: de Gruyter Studies in Mathematics Series , #5
  • Edition description: 2nd revised and extended ed. 2003
  • Edition number: 2
  • Pages: 572
  • Product dimensions: 6.69 (w) x 9.61 (h) x 1.25 (d)

Meet the Author

Gerhard Burde, Goethe University Frankfurt am Main, Germany; Heiner Zieschang †; Michael Heusener, Blaise Pascal University,France.
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Table of Contents

1 Knots and Isotopies 1
2 Geometric Concepts 15
3 Knot Groups 30
4 Commutator Subgroup of a Knot Group 52
5 Fibred Knots 68
6 A Characterization of Torus Knots 79
7 Factorization of Knots 91
8 Cyclic Coverings and Alexander Invariants 103
9 Free Differential Calculus and Alexander Matrices 125
10 Braids 142
11 Manifolds as Branched Coverings 172
12 Montesinos Links 189
13 Quadratic Forms of a Knot 219
14 Representations of Knot Groups 249
15 Knots, Knot Manifolds, and Knot Groups 282
16 The 2-variable skein polynomial 312
App. A Algebraic Theorems 325
App. B Theorems of 3-dimensional Topology 331
App. C: Tables 335
App. D Knot projections 0[subscript 1]-9[subscript 49] 363
Bibliography 367
List of Code Numbers 507
List of Authors According to Codes 509
Author Index 551
Glossary of Symbols 553
Subject Index 555
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