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

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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, supercoiling, etc.) and the action of various important life-sustaining enzymes (topoisomerases and recombinases). These experiments are often performed

IX X PREFACE on circular DNA molecules, in which changes in the geometric (supercoiling) or topological (knotting and linking) state of the DNA can be directly observed-witness the beautiful electron micrographs of DNA knots and links. The recently acquired ability to preform these experiments provides a chal• lenge for mathematics-to build mathematical models for DNA structure and enzyme action which explain the experimental observations (both qualita• tively and quantitatively), models which can be used to design further ex• periments and to predict the outcome of these experiments. Knot theory has also been involved in a recent fundamental and revolutionary theoreti• cal development, where the pioneering work of Vaughn F. R. Jones has set off an explosion of interaction between theoretical physics and mathematics. A whole new spectrum of invariants for 3-manifolds has been born. These new interactions between science and mathematics form a beautiful concrete example of, in the words of E. E. David, Jr, "the seemingly inevitable utility of mathematics conceived symbolically without reference to the real world ". I would like to thank Carole Kohanski and Jim Maxwell for their help in organizing this short course at the 1992 Baltimore AMS meeting, and to Donna Harmon for her help in the preparation of this volume. On behalf of the American Mathematical Society, I would like to acknowledge a grant from Genentech, Inc., sponsors of this AMS Short Course. Thanks also to the enthusiastic audience who attended the lectures; their interest, questions and comments were stimulating to all!

De Witt L. Sumners Index

achiral, 45 conceit, 137 Alexander polynomial, 131, 132, 134, 211, conformal field theory, 133, 139, 140, 186, 225 224 Alexander-Conway polynomial, 225 conformal quantum field theory, 172 algebraic topology, 131 connective constant, 77, 78 almost unknotted embedding, 82, 83 Conway, 132 almost unknotted graphs, 75 creations, 141 alternating diagram, 46 cross-channeling, 134 ambient isotopy, 42, 156 crossing, 135, 143, 187, 191 amplitude, 140, 141, 143, 144 crossing number, 45, 81 angular momentum, 195 cup, 187 angular momentum recoupling, 177 curvature, 223 annihilation of particles, 143 curvature tensor F, 222 annihilations, 141 antisymmetrizer, 204 determinant, 211 Artin braid group, 131, 139, 161 Dirac formalism, 143 atoms, 131 Dirac string trick, 146 direct repeats, 58 base pairs, 54 distributive recombination, 59 BFACF algorithm, 88 DNA on protein complexes, 25 binor calculus, 201 DNA repair, 6 bra-ket, 142 DNA replication, 4 bracket evaluation, 163 DNA topology, 1 bracket identity, 173 DNAse, 54 bracket polynomial, 153, 157, 169, 198 Dubrovnik polynomial, 198 braids, 161 bras, 142 elementary braid, 161 Burau representation, 215 Elliot-Biedenharn identity, 180, 181 energy, 168 cap, 187 entanglement complexity, 75, 81, 89, 91 categories, 145 entanglement number, 90 catenane, 2, 59, 99, 104 enzyme mechanism, 61 Chebyshev polynomial, 176 epsilon, 194 Chern-Simons Lagrangian, 140, 222 ether, 131 chiral, 45, 100 evolution of the topological structure of chirality, 99 DNA, 1 chromatic polynomial, 153, 170 classifying vector, 47 Feynman integrals, 133 collapse of the wave function, 144 Feynman path integral, 137, 141 combinatorial knot theory, 186 Four Color Theorem, 170, 172 combinatorial state model, 219 4-plat, 47 combinatorics, 133, 153 4-valent plane graph, 154 completeness, 143 Fox derivation, 214 complex numbers, 140 framed link, 220 composite, 45 framing, 226, 227 composition of morphisms, 188 free differential calculus, 211, 213, 215 247 248 INDEX

free fermion model, 211 knotting, 2 functional integral, 139, 186, 224 Kronecker delta, 192, 194 functors, 145 fundamental group, 134, 205, 211, 216 Lie algebras, 132, 137 ligase, 54 gauge equivalence, 140 light scattering, 87 gauge field theory, 222 link, 41 gauge potentials, 137, 140, 222 link crystal, 136 gel electrophoresis, 56, 88 link diagram, 43, 132, 143, 154, 167, 172 generalized Feynman path integral, 225 link invariant, 200 generalized polynomial, 213 linker DNA, 33 genetic sequence, 54 linker regions, 32 Grassmann variable, 211 linking, 18 linking number difference, 24 h, 30 linking number, 2, 18, 19, 20, 44 handlebody, 140 links with singularities, 146 Hecke algebra, 210 Lk(C,A), 18 helical repeat, 26, 30 local knots, 75 heuristic integral, 222 loop value, 173 higher dimensions, 146 Lorentz Transformation, 195 Hilbert space, 140, 142 holonomy, 137, 138 mass spectrometry, 116 holonomy of A, 222 mathematical physics, 132 Homfly polynomial, 198, 219, 225, 226 matrix elements, 191 Hopf algebra, 132, 199, 200, 201 matrix multiplication, 145 Matveev moves, 185 in vitro, 56 Matveev representation, 182 in vivo, 56 maxima, 143, 191 infinite cyclic cover, 211 medial construction, 156 integral, 201 medial link diagram, 156, 169 integral formalism, 220 Metropolis sampling, 85 integral heuristics, 219 minichromosome, 32 integral tangle, 50 minima, 143, 191 integrase, 57 minimal diagram, 46 interactions, 141, 143 mirror images, 131 inverted repeats, 59 Mobius ladder graphs, 101 Ising model, 132 Mobius ladders, 122 molecular Mobius band, 106 Jacobian matrix, 211 molecular trefoil knot, 97 Jones polynomial, 131, 137, 139, 145, 153, Monte Carlo, 85, 87 211, 216, 225, 226 morphism, 137, 145, 187, 188 Jones projector, 176 multiplicative relations, 164 Jordan Euler trail, 218 multivariable Alexander polynomial, 211

Kauffman polynomial, 225, 227 negative supercoiling, 10 ket-bras, 142 NMR, 117 kets, 142 nonstandard set theory, 231 knot, 41 normal vector field, 220 knot epistemology, 228 nih Vassiliev invariant, 219 knot polynomial, 81 nucleic acid, 1 knot probability, 76, 85 nucleosome, 25 knot theory, 131, 132, 143, 153 knot type, 42 one-dimensional genetic code, 5 knot-set, 230 operand, 136 knotted, 133 operator, 136 knotted arc, 79, 90 operator algebras, 132 knotted ball pair, 79, 80 orientation preserving, 42 knottedness, 134 oriented, 42 INDEX 249 original Jones polynomial, 161 relaxed DNA, 54 orthogonality identity, 181 Reshetikhin-Turaev invariant, 185, 186 outside bound tangle Of,, 61 ribbon knot, 212, 213 outside free tangle Of, 61 Riemann surface, 139, 140 outside tangle 0, 62 ring polymers, 74 RNA, 2 parental tangle P, 61 parity, 51 scattering amplitude, 195 particle interaction, 135 scattering matrix, 135 partition function, 133, 134, 168, 169, Seifert pairing, 211 186, 205 self reference, 232 pattern theorem, 78, 90 self-avoiding polygon, 74, 78 Penrose Binor Calculus, 194 self-avoiding walk, 74, 76, 78, 89 Penrose spin network, 198, 201 set theory, 229 permutation, 174, 186, 204 site-specific recombination, 11 physical level of rigor, 224 skein, 227 physical state, 134 skein calculation, 211, 227 physics, 131 skein identity, 219 pivot algorithm, 76, 85 skein polynomial, 132, 225 plane graph, 156 skein relation, 221 plane graph G, 170 skein theoretic, 137 plat closure, 166, 167 SL(2), 194 plat trace, 170 SL(2) quantum group, 187 Platonic, 132 SL(2)equation, 62 spin, 195 projected writhing number, 20 spin network, 172, 180, 205 spinor, 201 q-6j symbol, 178, 180, 184 state, 158, 191 #-spin network, 172 state summation model, 153 g-spin recoupling theory, 176 state transitions, 134 #-state Potts model, 168 states of a braid, 162 4-symmetrizer, 172, 173, 180 statistical mechanics, 132, 153, 211 quantum field theory, 133, 137, 153, 186 stereoisomers, 114 quantum gravity, 205 strand closure, 163 quantum group, 145, 153, 186, 198, 200 substrate, 57 quantum knots, 143 substrate equation, 62 quantum mechanics, 140, 144 supercoil, 54 quantum state, 142 supercoiled, 17 supercoiling, 24, 39 radius of gyration, 87 superhelical stress, 9 rapidity, 195, 196 surface linking number, 25, 26 rapidity parameter, 195 surface of a protein, 25 rational tangles, 50 SU(2), 139 recombinant tangle R, 61 SV40, 25, 34 recombination equations, 62 SV40 molecule, 24 recombination site, 57 symmetric group, 173 recursive definitions, 132 symmetry presentation, 103 recursive form, 231 synapsis, 57 Reidemeister move, 88, 135, 145, 155, 21 synaptic complex, 57 Reidemeister torsion, 186, 187, 212, 224 synaptosome, 57 regular isotopy, 219 250 INDEX tangle, 48, 162, 166, 186 Turaev-Viro partition function, 186 tangle calculus, 52 Tw, 22 tangle sum, 51 Tw(C,A), 22 tangle type, 49 twist, 18, 20, 22 tempered braid, 165, 167 twist move, 189 Temperley-Lieb algebra, 132, 143, 153, twist of C about A, 35 164, 168, 170, 172, 186 Temperley-Lieb elements, 175 uniform embedding, 84 tetrahedral symbol, 183 unoriented, 42 tetrahedron, 180 theta-curve, 125 Vac, 22 6-symbol, 178, 184 vacuum, 144 third dimension, 136 vacuum-vacuum amplitude, 145 3-manifold, 186 Varasoro algebra, 172 three-dimensional topology, 186 Vassiliev invariant, 216 three-dimensional manifold, 133 vertex, 135 three-vertex, 177 vertex weights, 158 Tn3 resolvase, 60 von Neumann algebra, 131 topoisomerase, 4, 54 vortices, 131 topological approach to enzymology, 40, 55 weaving pattern, 161 topological bracket polynomial, 159 Wess-Zumino-Witten, 139 topological deformation, 134 winding number, 25, 26 topological phenomenology, 146 Wirtinger presentation, 215 topological quantum field theories, 140 Witten functional integral, 222 topological stereoisomers, 116 Wr, 20, 21 topological string, 146 writhe, 18, 20, 160 topological symmetries, 125 topological symmetry group, 118 X-ray crystallography, 119 topologically chiral, 101, 114 topology, 41, 131 Yang-Baxter equation, 132, 187, 189, 193, transposase, 57 195, 209 transposition, 174 Yang-Baxter model, 153, 187, 198 trefoil knot, 133, 154 Yang-Baxter solution, 211, 215 triangulation, 133, 186 Turaev-Viro invariant, 153, 172, 182, 184, zero temperature limit, 169 185, 205, 224 Other Titles in This Series (Continued from the front of this publication)

21 P. D. Lax, editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976) 20 J. P. LaSalle, editor, The influence of computing on mathematical research and education (University of Montana, August 1973) 19 J. T. Schwartz, editor, Mathematical aspects of computer science (New York City, April 1966) 18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965) 17 R. Finn, editor, Applications of nonlinear partial differential equations in mathematical physics (New York City, April 1964) 16 R. Bellman, editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963) 15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962) 14 R. Bellman, editor, Mathematical problems in the biological sciences (New York City, April 1961) 13 R. Bellman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New York City, April 1960) 12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York City, April 1960) 11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April 1959) 10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University, April 1957) 9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, editor, Calculus of variations and its applications (University of Chicago, April 1956) 7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April 1955) 6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953) 5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of Technology, June 1952) 4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951) 3 R. V. Churchill, editor, Elasticity (University of Michigan, June 1949) 2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948) 1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University, August 1947)

(See the AMS catalog for earlier titles)