Knots and links in three-dimensional flows.

*(English)*Zbl 0869.58044
Lecture Notes in Mathematics. 1654. Berlin: Springer. x, 208 p. (1997).

Periodic orbits of three-dimensional flows may appear knotted and linked. The basic tools in this area of research come from two well-developed fields: template theory and dynamical systems theory. This book brings them together with an original presentation and is written in a very comprehensive and elegant style which I find will appeal to specialists in both fields.

Many interesting questions arise naturally in this context. Given a flow one would like to describe the knot and link types that are found among its periodic orbits. How does the “richness” of the dynamics of Smale flows in contrast to Morse-Smale flows reflect on the families of links that appear among periodic orbits of these flows? Do systems with conservation laws or symmetry properties support preferred families of links? Is the complexity of knotting related to topological entropy? This book provides insight and answers to these basic questions.

The book is essentially self-contained, although some familiarity with low-dimensional topology and dynamical systems is helpful. The background material for subsequent chapters is sketched in the first two chapters. The authors review basics from knot theory and dynamical systems theory in Chapter 1. In Chapter 2, the reader is introduced to the major tool: the template. A template is a branched two-manifold which in some sense supports the periodic orbits of a flow. A “template calculus”, a symbolic language that permits to manipulate templates and relations among them is developed. General results on templates, knots and links are obtained in Chapter 3. Chapter 4 addresses dynamical systems questions of the type we mentioned above. Also, the question how the knotting and linking data restricts the families of periodic orbits and the bifurcations which they undergo, is treated. Bifurcation invariants are established. Chapter 5 and 6 are directed towards researchers in these areas and describe the current state of affairs together with a list of open problems.

Many interesting questions arise naturally in this context. Given a flow one would like to describe the knot and link types that are found among its periodic orbits. How does the “richness” of the dynamics of Smale flows in contrast to Morse-Smale flows reflect on the families of links that appear among periodic orbits of these flows? Do systems with conservation laws or symmetry properties support preferred families of links? Is the complexity of knotting related to topological entropy? This book provides insight and answers to these basic questions.

The book is essentially self-contained, although some familiarity with low-dimensional topology and dynamical systems is helpful. The background material for subsequent chapters is sketched in the first two chapters. The authors review basics from knot theory and dynamical systems theory in Chapter 1. In Chapter 2, the reader is introduced to the major tool: the template. A template is a branched two-manifold which in some sense supports the periodic orbits of a flow. A “template calculus”, a symbolic language that permits to manipulate templates and relations among them is developed. General results on templates, knots and links are obtained in Chapter 3. Chapter 4 addresses dynamical systems questions of the type we mentioned above. Also, the question how the knotting and linking data restricts the families of periodic orbits and the bifurcations which they undergo, is treated. Bifurcation invariants are established. Chapter 5 and 6 are directed towards researchers in these areas and describe the current state of affairs together with a list of open problems.

Reviewer: K.Resende (Campinas)

##### MSC:

37C10 | Dynamics induced by flows and semiflows |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

34C25 | Periodic solutions to ordinary differential equations |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |