HANDBOOK OF KNOT THEORY ThisPageIntentionallyLeftBlank HANDBOOK OF KNOT THEORY Editors WILLIAM MENASCO Department of Mathematics University at Buffalo USA MORWEN THISTLETHWAITE Department of Mathematics University of Tennessee USA 2005 Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo ELSEVIER B.V. ELSEVIER Inc. ELSEVIER Ltd ELSEVIER Ltd Radarweg 29 525 B Street, Suite 1900 The Boulevard, Langford Lane 84 Theobalds Road P.O. Box 211, 1000 AE San Diego, CA 92101-4495 Kidlington, Oxford OX5 1GB London WC1X 8RR Amsterdam, The Netherlands USA UK UK q 2005 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. 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British Library Cataloguing in Publication Data Handbook of knot theory 1.Knot theory I.Menasco, William W., 1954- II.Thistlethwaite, Morwen 514.2’242 ISBN: 0-444-51452-X W1 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands Preface Accessible to the layman, with immense raw geometric appeal, yet in close touch with some elevated disciplines, knot theory holds a special place in mathematics. From its humble beginnings as an empirical science in the mid-19th century, the subject has undergone an extraordinary evolution, greatly accelerated over the last 25 years. One of the first serious researchers in knot theory, the notable Scottish physicist Peter Guthrie Tait, began his celebrated tabulation of knots of small crossing-number with the hope of elucidating the Kelvin vortex theory of atoms, but soon became entranced with the subject in its own right. He observed, perceptively, that he could not ever prove that two knots were inequivalent; indeed, that had to wait until the advent of Poincare´’s Fundamental Group in the early 1900s. Knot theorists now have a formidable arsenal of techniques for distinguishing knot and link types, but such considerations form only a narrow part of contemporary knot theory. Knots have found themselves involved with almost every major advance in low-dimensional topology, be it Papakyriakopoulos’s proof of Dehn’s Lemma in 1957 (whose original aim was to show that the group of a non-trivial knot could not be isomorphic to the integers), or Riley’s discovery in the 1970s of a hyperbolic structure on the complement of the figure-eight knot and Thurston’s subsequent profound work on geometric structures on 3-manifolds. Of equal importance are interactions with piecewise-linear 3-dimensional topology, with braids, with contact structures on manifolds, and with 4-dimensional topology, both classical and gauge-theoretical. It would be audacious to attempt even a summary of the full ramifications of knot theory, but mention must be made of Jones’s astonishing discovery in 1984 of an entirely new breed of polynomial invariants, which strengthened knot theory’s connection with the theory of braids and forged a completely new relationship with the mathematics of quantum field theory. In this volume, we present a collection of survey articles by leading experts on a cross- section of present-day knot theory. For us, reading these articles and witnessing the vast range of knowledge contained therein was an inspiring and humbling experience. We are grateful for this opportunity to thank the contributors for their sheer hard work and their willingness to share this knowledge. We would also like to thank the staff of Elsevier for their copious help with this project. William Menasco Morwen Thistlethwaite vv ThisPageIntentionallyLeftBlank List of Contributors Prof. Adams, C., Bronfman Science Center, Department of Mathematics, Williams College, Williamstown, MA 01267, USA, [email protected] (Ch. 1) Prof. Birman, J.S., Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027, USA, [email protected] (Ch. 2) Prof. Brendle, T.E., Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA, [email protected] (Ch. 2) Prof. Etnyre, J.B., University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA, [email protected] (Ch. 3) Prof. Friedman, G., Department of Mathematics, Yale University, 10 Hillhouse Ave/ P.O. Box 208283, New Haven, CT 06520-8283, USA, [email protected] (Ch. 4) Prof. Hoste, J., Pitzer College, Department of Mathematics, 1050 N Mills Avenue, Claremont, CA 91711, USA, [email protected] (Ch. 5) Prof. Kauffman, L.H., Department of Mathematics, Statistics and Computer Science, University of Illinois, 851 South Morgan Street, Chicago, IL 60607-7045, USA, [email protected] (Ch. 6) Prof. Livingston, C., Department of Mathematics, Indiana University, Bloomington, IN 47405, USA, [email protected] (Ch. 7) Prof. Menasco, W., Department of Mathematics, University at Buffalo, USA (Editor) Prof. Rudolph, L., Department of Mathematics and Computer Science and Department of Psychology, Clark University, Worcester, MA 01610, USA, [email protected] (Ch. 8) Prof. Scharlemann, M., Department of Mathematics, South Hall, University of California, Santa Barbara, CA 93106, USA, [email protected] (Ch. 9) Prof. Thistlethwaite, M., Department of Mathematics, University of Tennessee, USA (Editor) Prof. Weeks, J., www.geometrygames.org/contact.html (Ch. 10) viivii ThisPageIntentionallyLeftBlank Contents Preface v List of Contributors vii 1. Hyperbolic Knots 1 C. Adams 2. Braids: A Survey 19 J.S. Birman and T.E. Brendle 3. Legendrian and Transversal Knots 105 J.B. Etnyre 4. Knot Spinning 187 G. Friedman 5. The Enumeration and Classification of Knots and Links 209 J. Hoste 6. Knot Diagrammatics 233 L.H. Kauffman 7. A Survey of Classical Knot Concordance 319 C. Livingston 8. Knot Theory of Complex Plane Curves 349 L. Rudolph 9. Thin Position in the Theory of Classical Knots 429 M. Scharlemann 10. Computation of Hyperbolic Structures in Knot Theory 461 J. Weeks Author Index 481 Subject Index 483 ixix ThisPageIntentionallyLeftBlank CHAPTER 1 Hyperbolic Knots Colin Adams Bronfman Science Center, Department of Mathematics, Williams College, Williamstown, MA 01267, USA E-mail: [email protected] Contents 1. Introduction ...................................................... 3 2. What knot and link complements are known to be hyperbolic? . 3 3. Volumes of knots . ................................................ 8 4. Cusps .......................................................... 10 5. Meridians and other cusp invariants . ....................................... 11 6. Geodesics and totally geodesic surfaces . .................................. 13 Acknowledgements . ................................................ 16 References . ...................................................... 16 HANDBOOK OF KNOT THEORY Edited by William Menasco and Morwen Thistlethwaite q 2005 Elsevier B.V. All rights reserved 1