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http://dx.doi.org/10.1090/surv/095 Titles in This Series 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, Trie backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Mathematical Surveys and Monographs Volume 95 Braid and Knot Theory in Dimension Four Seiichi Kamada American Mathematical Society WDED* Editorial Board Peter Landweber Tudor Ratiu Michael Loss, Chair J. T. Stafford 2000 Mathematics Subject Classification. Primary 57Q45; Secondary 20F36, 57M05, 57M12, 57M25, 57Q35. ABSTRACT. Braid theory and knot theory are related to each other via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus one can use braid theory to study knot theory, and vice versa. In this book we generalize braid theory to dimension four. We develop the theory of surface braids and apply it to study surface links. Especially, the generalized Alexander and Markov theorems in dimension four are given. This book is the first place that contains a complete proof of the generalized Markov theorem. Surface links are also studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. A table of knotted surfaces is included with a computation of Alexander polynomials. The braid techniques are extended to represent link homotopy classes. Library of Congress Cataloging-in-Publication Data Kamada, Seiichi, 1964- Braid and knot theory in dimension four / Seiichi Kamada. p. cm. — (Mathematical surveys and monographs ; v. 95) Includes bibliographical references and index. ISBN 0-8218-2969-6 (alk. paper) 1. Braid theory. 2. Knot theory. I. Title. II. Mathematical surveys and monographs ; no. 95. QA612.23.K36 2002 514/.224—dc21 2002018274 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents Preface xi Chapter 0. Basic Notions and Notation 1 0.1. Properness and Local Flatness 1 0.2. Isotopy 2 0.3. Ambient Isotopy 3 0.4. Notation 3 Part 1. Classical Braids and Links 5 Chapter 1. Braids 7 1.1. Geometric Braids and the Braid Group 7 1.2. Configuration Space 8 1.3. Pure Braids and Braid Permutation 10 1.4. Brick Regular Neighborhood 11 1.5. Braid Isotopy Extension Theorem 12 1.6. Polygonal Braids 13 1.7. Braid Diagrams 14 1.8. Presentation of the Braid Group 15 Chapter 2. Braid Automorphisms 19 2.1. The Isotopies Associated with a Braid 19 2.2. Mapping Class Group 20 2.3. Hurwitz Arc System 21 2.4. Hurwitz Arc System of a Braid 22 2.5. Injectivity of $ 23 2.6. Artin's Braid Automorphism 23 2.7. Slide Action of the Braid Group 25 Chapter 3. Classical Links 27 3.1. Knots and Links 27 3.2. Basic Symmetries 28 3.3. The Regular Neighborhood of a Link 28 3.4. Trivial Links 29 3.5. Split Union and Connected Sum 30 3.6. Combinatorial Equivalence 30 3.7. Regular Projections 31 3.8. Link Diagrams 32 3.9. Reidemeister Moves 33 3.10. The Group of a Link 35 vi CONTENTS 3.11. A Note on Knots as Embeddings 38 Chapter 4. Braid Presentation of Links 41 4.1. Presenting Links by Braids 41 4.2. The Braiding Process 42 4.3. Markov's Theorem 43 Chapter 5. Deformation Chain and Markov's Theorem 47 5.1. £ Operation 47 5.2. Deformation Chains 48 5.3. Deformation Chains at the Triangulation Level 49 5.4. Height Reduction and Markov's Theorem 50 5.5. Notes 50 Part 2. Surface Knots and Links 51 Chapter 6. Surface Links 53 6.1. Surface Links 53 6.2. Trivial Surface Links 53 6.3. Combinatorial Equivalence 55 Chapter 7. Surface Link Diagrams 57 7.1. Generic Maps 57 7.2. Surface Link Diagrams 58 7.3. Elementary Moves to Diagrams 59 Chapter 8. Motion Pictures 63 8.1. Motion Pictures 63 8.2. Motion Pictures of Surface Links 63 8.3. The Motion Picture of a Non-locally Flat Surface 65 8.4. Elementary Critical Bands 66 8.5. Trivial Disk Systems 67 8.6. Proof of Lemma 8.7 69 Chapter 9. Normal Forms of Surface Links 71 9.1. Link Transformation Sequence 71 9.2. The Realizing Surface 72 9.3. Technical Lemmas on Realizing Surfaces 72 9.4. Closed Realizing Surface 75 9.5. The Normal Form 75 Chapter 10. Examples (Spinning) 79 10.1. Spinning Construction 79 10.2. Twist-Spinning Construction 80 10.3. Deform-Spinning Construction 82 10.4. Deformations 83 10.5. Deform-Spun Projective Planes 84 Chapter 11. Ribbon Surface Links 87 11.1. 1-Handle and 2-Handle Surgery 87 11.2. Ribbon Surface Links 88 CONTENTS vii 11.3. Slice versus Ribbon 89 Chapter 12. Presentations of Surface Link Groups 91 12.1. The Presentation from a Diagram 91 12.2. The Presentation from a Motion Picture 92 12.3. The Elementary Ideals 94 Part 3. Surface Braids 97 Chapter 13. Branched Coverings 99 13.1. Branched Coverings 99 13.2. Types of Branch Points 100 13.3. Riemann-Hurwitz Formula 101 13.4. Monodromy 102 13.5. Simple Branched Coverings 103 Chapter 14. Surface Braids 105 14.1. Surface Braids 105 14.2. Motion Pictures 105 14.3. Trivial Surface Braids 107 14.4. Simple Surface Braids 108 14.5. Equivalence Relations 109 Chapter 15. Products of Surface Braids 113 15.1. Products of Motion Pictures 113 15.2. Products of Surface Braids 114 15.3. The Surface Braid Monoid 114 Chapter 16. Braided Surfaces 117 16.1. Braided Surfaces 117 16.2. The Motion Picture of a Braided Surface 117 16.3. Equivalence Relations on Braided Surfaces 117 16.4.
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