de Gruyter Studies in Mathematics 5 Editors: Carlos Kenig · Andrew Ranicki · Michael Röckner de Gruyter Studies in Mathematics 1 Riemannian Geometry, 2nd rev. ed., Wilhelm P. A. Klingenberg 2 Semimartingales, Michel Me´tivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5Knots, Gerhard Burde and Heiner Zieschang 6Ergodic Theorems, Ulrich Krengel 7Mathematical Theory of Statistics, Helmut Strasser 8Transformation Groups, Tammo tom Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve´ 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev 19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima, Yoichi Oshima and Masayoshi Takeda 20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R. Bloom and Herbert Heyer 21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov 22 Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii, Victor E. Shatalov and Boris Yu. Sternin 23 Probability Theory, Heinz Bauer 24 Variational Methods for Potential Operator Equations, Jan Chabrowski 25 The Structure of Compact Groups, Karl H. Hofmann and Sidney A. Morris 26 Measure and Integration Theory, Heinz Bauer 27 Stochastic Finance, Hans Föllmer and Alexander Schied 28 Painleve´ Differential Equations in the Complex Plane, Valerii I. Gromak, Ilpo Laine and Shun Shimomura 29 Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel and Jakob Nielsen Gerhard Burde · Heiner Zieschang Knots Second Revised and Extended Edition Walter de Gruyter ≥ Berlin · New York 2003 Authors Gerhard Burde Heiner Zieschang Fachbereich Mathematik (Fach 187) Fakultät für Mathematik Universität Frankfurt am Main Ruhr-Universität Bochum Robert-Mayer-Str. 6Ϫ10 Universitätsstr. 150 60325 Frankfurt/Main 44801 Bochum Germany Germany Series Editors Carlos E. Kenig Andrew Ranicki Michael Röckner Department of Mathematics Department of Mathematics Fakultät für Mathematik University of Chicago University of Edinburgh Universität Bielefeld 5734 University Ave Mayfield Road Universitätsstraße 25 Chicago, IL 60637, USA Edinburgh EH9 3JZ, Scotland 33615 Bielefeld, Germany Mathematics Subject Classification 2000: 57-02; 57M25, 20F34, 20F36 Keywords: knots; links; fibred knots; torus knots; factorization; braids; branched coverings; Montesinos links; knot groups; Seifert surfaces; Alexander polynomials; Seifert matrices; cyclic periods of knots; homfly polynomial With 184 figures Țȍ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data Burde, Gerhard, 1931Ϫ Knots / Gerhard Burde, Heiner Zieschang. Ϫ 2nd rev. and extended ed. p. cm. Ϫ (De Gruyter studies in mathematics ; 5) Includes bibliographical references and index. ISBN 3 11 017005 1 (cloth : alk. paper) I. Zieschang, Heiner. II. Title. III. Series. QA612.2 .B87 2002 514Ј.224Ϫdc21 2002034764 ISBN 3 11 017005 1 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at Ͻhttp://dnb.ddb.deϾ. Ą Copyright 2003 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Preface to the First Edition The phenomenon of a knot is a fundamental experience in our perception of three dimensional space. What is special about knots is that they represent a truly intrinsic and essential quality of 3-space accessible to intuitive understanding. No arbitrariness like the choice of a metric mars the nature of a knot–atrefoil knot will be universally recognizable wherever the basic geometric conditions of our world exist. (One is tempted to propose it as an emblem of our universe.) There is no doubt that knots hold an important – if not crucial – position in the theory of 3-dimensional manifolds. As a subject for a mathematical textbook they serve a double purpose. They are excellent introductory material to geometric and algebraic topology, helping to understand problems and to recognize obstructions in this field. On the other hand they present themselves as ready and copious test material for the application of various concepts and theorems in topology. The first nine chapters (excepting the sixth) treat standard material of classical knot theory. The remaining chapters are devoted to more or less special topics depending on the interest and taste of the authors and what they believed to be essential and alive. The subjects might, of course, have been selected quite differently from the abundant wealth of publications in knot theory during the last decades. We have stuck throughout this book mainly to traditional topics of classical knot theory. Links have been included where they come in naturally. Higher-dimensional knot theory has been completely left out – even where it has a bearing on 3-dimensional knots such as slice knots. The theme of surgery has been rather neglected – excepting Chapter 15. Wild knots and Algebraic knots are merely mentioned. This book may be read by students with a basic knowledge in algebraic topology –atleast the first four chapters will present no serious difficulties to them. As the book proceeds certain fundamental results on 3-manifolds are used – such as the Papakyriakopoulos theorems. The theorems are stated in Appendix B and references are given where proofs may be found. There seemed to be no point in adding another presentation of these things. The reader who is not familiar with these theorems is, however, well advised to interrupt the reading to study them. At some places the theory of surfaces is needed – several results of Nielsen are applied. Proofs of these may be read in [ZVC 1980], but taking them for granted will not seriously impair the understanding of this book. Whenever possible we have given complete and self- contained proofs at the most elementary level possible. To do this we occasionally refrained from applying a general theorem but gave a simpler proof for the special case in hand. There are, of course, many pertinent and interesting facts in knot theory – especially in its recent development – that were definitely beyond the scope of such a textbook. To be complete – even in a special field such as knots – is impossible today and was not aimed at. We tried to keep up with important contributions in our bibliography. vi Preface to the First Edition There are not many textbooks on knots. Reidemeister’s “Knotentheorie” was con- ceived for a different purpose and level; Neuwirth’s book “Knot Groups” and Hillman’s monograph “Alexander Ideals of Links” have a more specialised and algebraic interest in mind. In writing this book we had, however, to take into consideration Rolfsen’s remarkable book “Knots and Links”. We tried to avoid overlappings in the contents and the manner of presentation. In particular, we thought it futile to produce another set of drawings of knots and links up to ten crossings – or even more. They can – in perfect beauty – be viewed in Rolfsen’s book. Knots with less than ten crossings have been added in Appendix D as a minimum of ready illustrative material. The tables of knot invariants have also been devised in a way which offers at least something new. Figures are plentiful because we think them necessary and hope them to be helpful. Finally we wish to express our gratitude to Colin Maclachlan who read the manu- script and expurgated it from the grosser lapsus linguae (this sentence was composed without his supervision). We are indebted to U. Lüdicke and G. Wenzel who wrote the computer programs and carried out the computations of a major part of the knot invariants listed in the tables. We are grateful to U. Dederek-Breuer who wrote the program for filing and sorting the bibliography. We also want to thank Mrs. A. Huck and Mrs. M. Schwarz for patiently typing, re-typing, correcting and re-correcting abominable manuscripts. Frankfurt (Main)/Bochum, Summer 1985 Gerhard Burde Heiner Zieschang Preface to the Second Edition The text has been revised, some mistakes have been eliminated and Chapter 15 has been brought up to date, especially taking into account the Gordon–Luecke Theorem on knot complements, although we have not included a proof. Chapter 16 was added, presenting an introduction to the HOMFLYpolynomial, and including a self-contained account of the fundamental facts about Hecke algebras. A proof of Markov’s theorem was added in Chapter 10 on braids. We also tried to bring