Knots and links in three-dimensional flows by, Ghrist, Homles and Sullivan

Here is the introduction:

This book concerns knots and links in dynamical systems.

Knot and link theory is an appealing subject. The basic ideas and results may be appreciated intuitively, simply by playing with pieces of string (\eg \cite{Ash44,Ada94}). Nonetheless, in spite of seafarers' development of sophisticated knots over thousands of years, the mathematical theory of knots began only in the nineteenth century. Its origins lie in Gauss's interest in electromagnetic field lines \cite{Gau77} and in attempts to classify knotted strings in the \ae ther, which Lord Kelvin and others thought might correspond to different chemical elements \cite{Tho69,Tai98}. It rapidly shed its physical origins and became a cornerstone of low-dimensional topology.

The roots of dynamical systems theory are considerably older and more tangled; they may be found in the {\em Principia Mathematica} of Isaac Newton and in attempts to model the motions of heavenly bodies. {\em Ab initio\/} the subject requires more technical apparatus: the differential and integral calculus, for a start; but at the same time it has kept closer touch with its physical origins. Moreover, in the last hundred years, it too has (re)acquired a strong geometrical flavor. In fact it was in an assault on the (restricted) three body problem of celestial mechanics \cite{Poi90}, in response to the prize competition to celebrate the 60th birthday of King Oscar II of Sweden and Norway, that Henri Poincar\'e essentially invented the modern, geometric theory of dynamical systems. He went on to develop his ideas in considerable detail in {\em Nouvelles Methodes de la M\'ecanique Celeste} \cite{Poi99}. Today, following this work, that of the Soviet school, including Pontriagin, Andronov, Kolmogorov, Anosov, and Arnol'd, and of Moser and Smale and their students in the West, the subject has reached a certain maturity. Over the last twenty years, it has escaped from Mathematics Departments into the scientific world at large, and in its somewhat ill-defined incarnations as chaos theory'' and nonlinear science,'' the methods and ideas of dynamical systems theory are finding broad application.

The basic world of a dynamical system is its {\em state space\/}: a (smooth) manifold, $M$, which constitutes all possible states of the system, and a mapping or flow defined on $M$. In one of our principal motivating examples, systems of first order ordinary differential equations (ODEs), the vector field thus specified generates a flow $\phi_t:M \ra M, t\in\real$. The general problem tackled by dynamical systems theorists is to describe $\phi_t$ geometrically, via its action on subsets of $M$. This implies classification of the asymptotic behaviors of all possible solutions, by finding fixed points, periodic orbits and more exotic recurrent sets, as well as the orbits which flow into and out of them. In many applications $\phi_t$ also depends on external parameters, and the topological changes or {\em bifurcations\/} that occur in $M$ as these parameters are varied, are also of interest. In studying these and related phenomena, one abandons the fruitless search for closed form solutions in terms of elementary or special functions, and seeks instead qualitative information.

Over the past decade, knot theory, once in the inner sanctum of pure mathematics, has been leaking out into other fields through several successful applications. These range from molecular biology, involving topological structures of closed DNA strands \cite{Sum90}, to physics, led by surprising connections with statistical mechanics \cite{Kau91*} and quantum field theory \cite{Wit89,Ati90}. Likewise, over the past ten to fifteen years, several attempts have been made to draw knot theory and dynamical systems closer together. The key idea is simple: a closed (periodic) orbit in a three-dimensional flow is an embedding of the circle, $S^1$, into the three-manifold that constitutes the state space of the system, hence it is a knot. Similarly, a finite collection of periodic orbits defines a link.

Several natural questions immediately arise, directed at the following goal: given a flow, perhaps generated by the vector field of a specific ODE, describe the knot and link types to be found among its periodic orbits. Do nontrivial knots occur? How many distinct knot types are represented? How many of each type? Do well-known families, such as torus knots, algebraic knots, or rational tangles, appear in particular cases? In any cases? Are there new'' families of knots and links which arise naturally in certain flows? Do Hamiltonian and other systems with conservation laws or symmetries support preferred families of links? Do chaotic'' flows contain inherently richer knotting than simple (Morse-Smale) flows? Indeed, how complicated can things get? -- is there a single ODE among whose periodic orbits can be found representatives of {\em all} knots and links? Such questions might occur to topologists. Indeed, it was R.F. Williams, in the context of a seminar on turbulence conducted in the Mathematics Department at Berkeley in 1976, who first conjectured that nontrivial knotting occurs in a well-known set of ODEs called the Lorenz equations \cite{Wil77}.

Dynamicists, in contrast, might seek to use knot and link invariants to describe periodic orbits and so help them better understand the underlying ODEs. In a parametrised family of flows, for example, one can observe sequences of bifurcations in which a simple invariant set containing, say, one or two periodic orbits, grows'' into a chaotic set of great complexity, containing a countable infinity of periodic orbits. In many cases, the periodic orbits are dense in the set of interest; sometimes that set is a so-called strange attractor. The existence-uniqueness theorem for solutions of ODEs implies that, as periodic orbits deform under parameter variation, they cannot intersect or pass through one another. Knot and link types therefore provide topological invariants which may be attached to families of periodic orbits. Can such invariants be used to identify orbit genealogies -- to trace the bifurcation sequences in which they arose? (A favorite problem is to describe bifurcation sequences in the two-parameter family of maps introduced by H\'enon \cite{Hen76}, which provides a model for Smale's famous horseshoe map.) Can operations in which new knots are created from old, such as composition and cabling, be associated with specific local bifurcations? Is the complexity of knotting related to other measures of dynamical complexity, such as topological entropy? Does knot theory provide finer invariants than entropy for the classification of flows?

Of course, since periodic orbits form knots only in {\em three}-dimensional flows, applications to dynamical systems in general are severely limited. Nonetheless, many of the rich and wonderful behaviors that currently engage dynamicists are already manifest in three dimensions, and so it seems well worth applying whatever tools we can to this case. In any event, we hope the reader will find the subject as beautiful, and attractive, as we do.

The Contents of this Volume

This book attempts to bring together two largely disparate and well developed fields, which have thus far only met in the pages of specialised research journals. As such, it cannot substitute for a proper course or text in either field. Chapter 1, to follow immediately, provides a rapid review of the principal aspects of knot theory and dynamical systems theory required for the remainder of the book. In Chapter 2 we develop the major tool which allows us to pass back and forth between hyperbolic flows and knots: the {\em template}. This was introduced (under the name knot holder'') over twelve years ago in two papers of Birman and Williams \cite{BW83a,BW83b}. In dynamical systems it is common to use {\em Poincar\'e} or {\em return maps} to reduce a flow to a mapping on a manifold of one lower dimension. While Poincar\'e maps preserve certain periodic orbit data, information on how the orbits are embedded in the flow is lost. The template preserves that information, and likewise reduces dimension. In Chapter 2 we develop a host of related tools: subtemplates, template inflations and renormalisations, and the symbolic language which allows us to manipulate templates and explore relations among them. We also introduce some of the particular (families of) templates which will concern us later.

Equipped with our basic tools, in Chapter 3 we obtain some general results on template knots and links, including the facts that, while specific templates may not contain {\em all\/} knots and links, every template contains infinitely many distinct knot types. We then describe a {\em universal\/} template, which {\em does\/} contain all (tame) knots and links, and which, moreover, arises rather naturally in certain classes of structurally stable three dimensional flows. In the final section, we explore the embedding problem:'' the question of which templates can be embedded in other templates. By considering isotopic embeddings, we are able to recognise universal templates hidden in ostensibly simpler ones.

The fourth chapter concerns bifurcations and knots, and directly addresses the kinds of dynamical systems questions raised in our opening paragraphs. In particular we focus on specific templates related to the H\'enon mapping and the creation of horseshoes. Here, in contrast to the limitless riches of Chapter 3, there are severe restrictions on links (all crossings are of one sign), which lead to uniqueness results and order relations on orbit creation in local bifurcations. We also explore knot types born in certain global or homoclinic bifurcations, by lifting the contrast between dynamically simple and dynamically complex bifurcations to the knot-theoretic level. In so doing, we derive a rather general set of sufficient conditions for a third-order ODE to support all links as periodic orbits.

Chapter 5 returns to basic template theory and presents the current state of affairs in template classification and invariant theory. We commence with a discussion of what a sensible definition of template equivalence should be, based on intuition developed in Chapters 3 and 4, and continue with a primitive but useful invariant: a zeta-function for a restricted class of templates. This will be seen to relate nicely to the underlying symbolic dynamics, yielding an easily-computed invariant which encodes twisting'' information in the compact package of a rational function.

Throughout Chapters 2-5 we strive to present, for the first time, a fairly complete picture of the theory of templates. As such, we include key results of Franks, Birman, Williams and others, although we focus primarily on our own work, relegating to an appendix some related work beyond the immediate scope of this monograph. Accordingly, Appendix A contains brief reviews of work by Morgan, Wada, and others on nonsingular Morse-Smale flows on three-manifolds, which contain only limited classes of knots. This is then contrasted with the work of Franks and work in progress by Sullivan on nonsingular Smale flows on the three-sphere.

Despite the title, we in no way claim to include every major result in the overlap of dynamics and knot theory. In particular, there is a natural dichotomy between knots arising from suspended surface homeomorphisms and closed orbits in flows on three-manifolds: this text focuses on the latter situation. The forthcoming book by P. Boyland and T. Hall \cite{BH96} deals with the former --- there is a great deal of beautiful work being done in this area of Nielsen theory and braid types'' for surface automorphisms \cite{Boy94,Boy89}. In addition, knot theory intersects with dynamics in examining problems of integrable Hamiltonian systems \cite{FN91}, the existence of minimal flows on three-manifolds \cite{Gut95} and contact geometry \cite{EG96a}. Finally, analogues of knotting and linking for nonperiodic, minimal orbits \cite{BM95,MM80} and asymptotic'' linking of orbits \cite{GST95,GG96} are very exciting, particularly since there are applications to magnetohydrodynamics \cite{Arn89} and fluid mechanics \cite{Mof86}.