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New Scientific Applications of Geometry and Topology http://dx.doi.org/10.1090/psapm/045 Other Titles in This Series 51 Louis H. Kauffman, editor, The interface of knots and physics (San Francisco, California, January 1995) 50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January- 1994) 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and W7illiam Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. W. McKelvey, editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984) 31 B. Gopinath, editor, Computer communications (Denver, Colorado, January 1983) 30 Simon A. Levin, editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981) 28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982) 26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979) 23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January 1978) (Continued in the back of this publication) New Scientific Applications of Geometry and Topology AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics Proceedings of Symposia in APPLIED MATHEMATICS Volume 45 New Scientific Applications of Geometry and Topology De Witt L. Sumners, Editor Nicholas R. Cozzarelli Louis H. Kauffman Jonathan Simon De Witt L. Sumners James H. White Stuart G. Whittington Q| American Mathematical Society 1^ IIII11II/^ Providence, Rhode Island LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE NEW SCIENTIFIC APPLICATIONS OF GEOMETRY AND TOPOLOGY HELD IN BALTIMORE, MARYLAND JANUARY 6-7, 1992 The AMS Short Course Series is sponsored by the Society's Program Committee on National Meetings. The Series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings. 1991 Mathematics Subject Classification. Primary 53A05, 57M25; Secondary 82B20, 82B41, 82D60, 92C40, 92E10. Library of Congress Cataloging-in-Publication Data New scientific applications of geometry and topology / De Witt L. Sumners, editor; Nicholas R. Cozzarelli... [et al.]. p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 45. AMS short course lecture notes) The short course was held in Baltimore, Md., January 6-7, 1992. Includes bibliographical references and index. ISBN 0-8218-5502-6 (acid-free paper) 1. Geometry, Differential—Congresses. 2. Knot theory—Congresses. 3. Science— Mathematics—Congresses. I. Sumners, De Witt L. II. Cozzarelli, Nicholas R. III. American Mathematical Society. IV. Series: Proceedings of symposia in applied mathematics; v. 45. V. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. QA641.N42 1992 92-26335 516.3'6—dc20 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material 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-permission9math.ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1992 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. W Printed on recycled paper. Portions of this volume were printed directly from author-prepared copy. Portions of this volume were typeset by the authors using AMS-T^X, the American Mathematical Society's Tr*X macro system. Papers in this volume are final versions, which will not be published elsewhere. 10 9 8 7 6 5 4 3 2 00 99 98 97 96 95 Table of Contents Preface ix Evolution of DNA Topology: Implications for its Biological Roles NICHOLAS R. COZZARELLI 1 Geometry and Topology of DNA and DNA-Protein Interactions JAMES H. WHITE 17 Knot Theory and DNA DE WITT L. SUMNERS 39 Topology of Polymers STUART G. WHITTINGTON 73 Knots and Chemistry JONATHAN SIMON 97 Knots and Physics Louis H. KAUFFMAN 131 Index 247 Preface Geometry and topology are subjects generally considered to be "pure" mathematics. Both originated in the effort to describe and quantize shape and form in order to understand the "real" world. Both enjoy a robust and sustained internal intellectual life, abstracted from the "reality" of their ori• gins. Recently, some of the methods and results of geometry and topology have found new utility in both wet-lab and theoretical science. Conversely, science is influencing mathematics, from posing questions which call for the construction of mathematical models to the importation of theoretical meth• ods of attack on long-standing problems of mathematical interest. A case in point is the subject of knot theory, which is utilized to a greater or lesser degree in each of the six papers in this volume. Knot theory traces its mathematical origins to the work of Gauss on computing inductance of linked circular wires, and to the work of Kelvin and Tait on the vortex theory of atoms. Knot theory is the study of entanglement and symmetry of elas• tic graphs in 3-space. It has proven to be fundamental as a laboratory for the development of algebraic topology invariants and in the understanding of the topology of 3-manifolds. During the last decade, laboratory scientists have become increasingly aware that the analytical techniques of geometry and topology can be used in the interpretation and design of experiments. Chemists have long been interested in developing techniques that will allow them to synthesize molecules with interesting 3-dimensional structure (knots and links). Polymer scientists study the chemical and physical ramifications of random topological entanglement in large molecules. Models for molecu• lar structure must be built and understood; reactions which produce specific 3-dimensional shapes must be designed; chemical proof of structure must be produced, and these proofs often involve the use of topology to interpret data such as NMR (Nuclear Magnetic Resonance) spectra. Molecular biologists know that the spatial conformation of DNA and the proteins which act on DNA is vital to their biological function; moreover, differential geometry and knot theory can be used to describe and quantize the 3-dimensional struc• ture of DNA and protein-DNA complexes. Biologists devise experiments on circular DNA which elucidate 3-D molecular conformation (helical twist, su- percoiling, etc.) and the action of various important life-sustaining enzymes (topoisomerases
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