http://dx.doi.org/10.1090/surv/055 Selected Titles in This Series

55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued , 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of poly logarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baer^istein, David Drasin, Peter Duren, and Albert Marden, Editors, The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 (Continued in the back of this publication) Mathematical Surveys and Monographs Volume 55

Knotted Surfaces and Their Diagrams

J. Scott Carter Masahico Saito

American Mathematical Society Editorial Board Howard A. Masur Michael Renardy Tudor Stefan Ratiu, Chair

1991 Mathematics Subject Classification. Primary 57Q45.

ABSTRACT. We explore the diagrammatic theory of knotted surfaces in 4-dimensional space. The notion of a diagram is fully developed in several contexts. The theory of Reidemeister type moves is presented in each context. We outline the theory of surface braids. Several aspects of knotted surfaces that contrast them with the classical theory of knotted are presented. Methods of computing the and related invariants are given. Several generalizations of the Yang-Baxter equation are discussed, and we sketch the category theory of knotted surfaces.

Cover figure by J. Scott Carter and Masahico Saito from Chapter 1 of this volume.

Library of Congress Cataloging-in-Publication Data Carter, J. Scott. Knotted surfaces and their diagrams / J. Scott Carter, Masahico Saito. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 55) Includes bibliographical references and index. ISBN 0-8218-0593-2 (hardcover : alk. paper) 1. theory. 2. Surfaces. I. Saito, Masahico, 1959- . II. Title. III. Series: Mathemat­ ical surveys and monographs ; no. 55. QA612.2.C37 1998 514/.224—dc21 97-34494 CIP

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 (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org.

© 1998 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 03 02 01 00 99 98 To Huong and Sean, Shuko and Kaita Contents

Preface ix

Chapter 1. Diagrams of Knotted Surfaces 1 1.1. Classical knot diagrams 1 1.2. Knotted surface diagrams 2 1.3. Reidemeister moves of classical 12 1.4. Movies of knotted surfaces 14 1.5. Charts of knotted surfaces 18 1.6. Examples: how to draw charts and decker curves 20 1.7. Symbolic presentations of classical knots 33 1.8. Sentences of knotted surfaces 34 1.9. Other diagrammatic methods 38

Chapter 2. Moving Knotted Surfaces 41 2.1. Equivalence of knotted surfaces 41 2.2. Roseman moves 42 2.3. Movie moves 44 2.4. Chart moves 52 2.5. The grammar of knotted surfaces 75 2.6. Singularities of knotted surface isotopies 78 2.7. Coffee break 88

Chapter 3. Braid Theory in Dimension Four 97 3.1. Classical braid theory 97 3.2. Surface braids 99 3.3. Charts of surface braids 100 3.4. Braid movies 116 3.5. Moves for charts and braid movies 117 3.6. Homotopy interpretations 123

Chapter 4. Combinatorics of Knotted Surface Diagrams 131 4.1. Orientations of the double and triple decker set 131 4.2. Surfaces in 3-space that do not lift 133 4.3. Smoothing triple points 145 4.4. Normal Euler numbers and branch points 148 4.5. Formulas for colored triple points 161 4.6. Some combinatorics of charts and sentences 166

Chapter 5. The Fundamental Group and the Seifert Algorithm 169 5.1. Wirtinger presentations for classical knots 169 5.2. Wirtinger presentations for knotted surfaces 171 viii CONTENTS

5.3. The Alexander module 181 5.4. A Seifert algorithm for knotted surfaces 188 Chapter 6. Algebraic Structures Related to Knotted Surface Diagrams 203 6.1. Generalizations of the Yang-Baxter equation 203 6.2. Category theory of knotted surfaces 214 6.3. Conclusion 241 Bibliography 243

Index 257 Preface

The purpose of this book is to develop the diagrammatic theory of knotted surfaces in 4-dimensional space in analogy with the classical theory of knotted and linked in 3-space. This goal may sound unachievable to some readers, how can we perceive phenomena that occur in 4-space? A related issue is perception in 3-dimensional space which we discuss briefly. For sighted people, the 3-dimensional world is projected upon a 2-dimensional surface, the retina, at any particular moment. Spatial relations are determined by a pair of two-dimensional images. The sense of touch is patently two-dimensional since the world comes into contact with us by means of our skin — a 2-dimensional surface. Sound is perceived as vibrations along a 2-dimensional membrane. The sense of taste is discrete with four states that are either excited or not. Only the olfactory sense has a degree of multi-dimensionality; it is difficult to classify and relate various smells. Each odor seems to be an independent quantity. But the sense of smell may also be discrete with an enormously large set of states. Since we usually perceive the world by a series of 2-dimensional impression, how do we come up with a 3-dimensional model of it? One can argue that the relative position of the wrist, elbow, and shoulder allow for a 3-dimensional world. So even though the world projects to us on our skin, retinas, or eardrums, we see the world as 3-dimensional. Additional perceptual clues come from moving the eyes to see around a corner, moving the hands to feel a different facet, or turning the ear towards a sound. So in order to develop the diagrammatic theory of knotted surfaces in 4- dimensions, we will project the surfaces into 3-dimensions, and we will move the surfaces around to see their different facets. It is not unreasonable that we will develop some 4-dimensional intuition in the process. In classical , invariants (Alexander, Conway, Jones, HOMPTFLY, Kauffman polynomials) are computed diagrammatically. Category theoretical in­ terpretations of knot diagrams play a key role in the study of quantum invariants. The braid form of a classical knot, which is both algebraic and diagrammatic, is used not only to define new invariants but also as geometric machinery for the study of knots. Most of these concepts and computations can be generalized to 4-dimensions via diagrams. Thus we will develop the theory of knotted surfaces and thereby provide the machinery for algebraic and geometric computations. Here is the outline of the book. Chapter 1 develops the notion of a knotted surface diagram. A diagram consists of a generic surface in 3-space together with crossing information indicated along the double point curves. Such a diagram can be projected further onto a plane to create a chart — a planar graph with labeled edges. The chart can be used to reconstruct the surface and to construct two other models. A movie consists of the

ix x PREFACE diagram with a fixed height function in 3-space. In such a movie we can consistently fix the height functions in the stills to create a fully combinatorial description of the surface. The combinatorial description is called a sentence; this is a sequence of words that are related by some grammatical rules. We give examples of each of these descriptions, and discuss some other diagrammatical methods. Chapter 2 contains the theory of Reidemeister moves. For each description of the knotted surfaces there is a finite set of moves such that equivalent knottings are related by a finite sequence of moves taken from this set. We give examples of applications of the moves in the various context. Chapter 2 closes with the classical argument that a coffee cup and a doughnut represent isoptopic surfaces in 3-space. Chapter 3 reviews the theory of surface braids that has been extensively devel­ oped by S. Kamada. We discuss generalizations of Alexander, Markov and Artin theorems. In particular, for a generalized Artin theorem, we give a finite list of moves to surfaces braids such that equivalent surface braids are related by a finite application of moves taken from this list, and we give examples of the Alexander isotopy. We close the chapter with a discussion on a homotopy theory interpretation of the surface braids. Chapter 4 contains material that contrasts the knotted surface case with the classical theory. We show that not all generic surfaces lift to embeddings. Triple point smoothing and applications thereof are given. We define signs and colors of triple and branch points, and relate them to the normal Euler number. Cancellation of cusps and branch points on the projections are discussed. Some of the work in this chapter is joint work with Vera Carrara. Chapter 5 contains methods for computing the fundamental group and a pre­ sentation matrix for the Alexander module of the knotted surface. We give explicit computations for several examples. The chapter closes with a description of the Seifert algorithm for knotted surfaces. Such an algorithm was used by Giller [Gi] in the case that the projection had no triple points. We developed the full algorithm in [CS2] and constructed Heegaard diagrams using our algorithm; Kamada wrote a version of the algorithm in the surface braid case. We use Kamada's approach to give a Heegaard diagram of the Seifert solid in the case the surface is given in braid form. Chapter 6 is a review of the algebraic and categorical aspects of knotted sur­ faces. We present solutions to the equations that are generalizations of the Yang- Baxter equation. Our solutions are based on diagrammatic methods and provide a good testimony to the power of these methods. The definition of a braided monoidal 2-category with duals (as given in [KV2], [BN] and [BL]) is sketched. We indicate that embedded surfaces in 3-space form a monoidal 2-category with duals while surfaces embedded in 4-space form a braided monoidal 2-category with duals. The chapter closes with the result of Baez and Langford that states that embedded surfaces form a free braided monoidal category with duals on one self dual object generator. This result forms the backbone of the future search for invariants that are analogous to the Jones . Some of the exercises are labeled research problems. That means that we do not know the outcome of the research. If the reader finds a solution before we do, then that is great! Throughout this book, the term the classical case refers to the theories of knotted and linked circles in 3-dimensional space or planar diagrams thereof. All manifolds and maps are smooth, and 4-space has the standard smooth structure. PREFACE XI

Acknowledgements. Some material in this book was presented in mini-courses taught by JSC at the Instituto de Ciencias Matematica Sao Carlos, Universidade de Sao Paulo, and Instituto de Matematica e Estatistica, Universidade de Sao Paulo during 1996. His visit to Brasil was supported by a Faculty Service and Development Award from the University of South Alabama, and by grants from FAPESP, Brasil. Additional support for JSC was provided by a grant from the National Security Agency. The University of South Florida supported MS with a grant from the Research and Creative Scholarship Grant Program. We offer our gratitude to several people for valuable discussions and comments. First of all, we are grateful for our wives' and families' support and understanding as the project was being developed. We have benefitted from correspondence and comments of John Baez, Vera Carrara, Richard Hitt, Seiichi Kamada, Misha Kapra- nov, Louis Kauffman, Laurel Langford, Joachim Rieger, Maria "Cindinha" Ruas, Daniel Silver, James Stasheff, and Oleg Viro. The financial and moral support of Cameron Gordon and Kunio Murasugi was essential during the early development of this research. Sergei Gelfand and Sarah Donnelly at the AMS deserve special kudos for working with us. They have been patient with our writing quirks, and persistent in getting the manuscript to press.

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Index i/j-move, 34 colored triple points, 161 r-decker set, 4 counit, 215 2-braidings, 234 counit 2-morphism, 222 2-category, 219 counit of an object, 223 2-category, braided monoidal, 227 cross-cap, 5 2-category, braided monoidal with duals, 232 cusp, 14 2-category, monoidal, 220, 221 2-category, monoidal with duals, 222 decker set, 4 2-tangle, 232 Dehn presentation, 183 2-tangles, 232 depth of a knotted surface, 168 diagram, 9 width of a knotted surface, 167 distinguished region, 138 double decker set, 3, 131 adjoint, 223 double point set, 2 Alexander index, 183 dual of a 1-morphism, 222 Alexander invariants, 181 dual of a 2-morphism, 222 Alexander isotopy, 105 dual of an object, 223 Alexander modules, 181 Alexander polynomial, 182 EBC, 116 Artin's construction, 23 elementary braid changes (EBCs) , 116 axis, 78 elementary string interactions (ESIs), 15 equivalence of knotted surfaces, 41 black vertex, 100 ESI, 15 braid chart, 100 braid chart moves, 117 FESI, 37 braid group, 97 fold, 14 braid movie, 116 Frenkel-Moore equation, 205 braid movie moves , 122 full elementary string interaction (FESI), 37 braided monoidal 2-category, 227 Gauss code, 27, 170 braided monoidal 2-category with duals, 232 generated, 232 braiding, 215, 227, 233, 234 generic, 1 braiding, 1-braiding, 234 generic surface, 2 braiding, 2-braiding, 234 grammar of knotted surfaces, 75 branch point, 2 broken surface diagram, 4 height direction, 13 Burau representation, 187 height function, 13, 14 height of a knotted surface, 168 C-moves, 118 horizontal axis, 78 Carter, Albert, 36 horizontal composition, 219 chart, 19 hyperbolic splitting, 39 chart moves, 52 classical knot, 1 Kinoshita-Terasaka knot, 23 closed braid, 98 Klein bottle, 5 closed surface braid, 99 knot, 1 colorable decker curve, 133 knot diagram, 1 colored branch points, 161 knotted surface, 2 colored double curve, 161 knotting, 2

257 258 INDEX lift able surface, 133 unframed object, 232 link, 1 unit, 215 lower decker curve, 131 unit 2-morphism, 222 unit of an object, 223 Markov stabilization, 98 unitary 2-morphism, 223 middle sectional braid, 187 unknotted, 2 monoidal 2-category, 220, 221 unknotting conjecture, 173 monoidal 2-category with duals, 222 unliftable surface, 133 move, types, 43 upper decker curve, 131 movie description, 15 vertical axis, 78 normal Euler number, 148 vertical composition, 219 normal form, 39 white vertex, 100 order ideal, 182 Whitney's congruence, 150 Whitney's umbrella, 2 passing a branch point through a third , 169, 171 43 word, 35 permutohedron, 127, 210, 212, 229 pinch point, 2 Yang-Baxter, 217, 229 planar tetrahedral equation, 205 Yang-Baxter equation, 203 projective plane, 5 YBE, 203, 215, 217, 229 quadruple point move, 43 Zamolodchikov equation, 206 quantum invariants, 12 Zeeman's construction, 28 ray, 19 Reidemeister moves, 12 retinal plane, 18 ribbon knots, 20 Roseman moves, 42

Seifert algorithm, 188 Seifert shell, 190 sentence, 35 simple surface braid, 99 singular point, 4 smoothing triple points, 145 spun knots, 23 stevedore's knot, 23 surface braid, 99 surface braid index, 99 surface braids, 139, 165 symmetric word, 120 tangle, 217 tangle, 2-tangle, 232 Temperley-Lieb algebra, 205 tetrahedral move, 43 triagulator, 223 triple decker set, 3 triple point, 2 twist spun knots, 28 type-I bubble move, 43 type-I saddle move, 43 type-I/type-I-inverse move, 43 type-II bubble move, 43 type-II saddle move, 43 type-II/type-II-inverse move, 43 type-III/type-III-inverse move, 43 types of moves, 43 Selected Titles in This Series (Continued from the front of this publication)

20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964 7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961 6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N. Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D. Tamarkin, The problem of moments, 1943