Joint with Robert Ghrist and Phil Holmes. Springer-Verlag, 1997. 208 pages.
Abstract. In an earlier paper the second author made a study of the knotted periodic orbits in a strange attractor for a set of differential equations in a paper by Clark Robinson. The attractor is modeled by a Lorenz-like template. It was shown that the knots and links are positive but need not be positive braids. Here we show that they are fibered, have positive signature, and that each knot-type appears infinitely often. We then construct a zeta type function that counts periodic orbits by the twisting of the local stable manifolds.
Abstract. This is a survey article submitted by invitation for inclusion in the Encyclopedia of Knot Theory, CRC Press. Published on December 2, 2020.
Abstract. In this paper, we discuss how to realize a non singular Smale flow with a four band template on 3-sphere. This extends the work done by the second author on Lorenz Smale flows, Bin Yu on realizing Lorenz Like Smale flows on 3-manifold and continues the work of Elizabeth Haynes and the second author on realizing simple Smale flows with a different four band template on 3-sphere.
Abstract. Smale flows on 3-manifolds can have invariant saddle sets that are suspensions of shifts of finite type. We look at Smale flows with chain recurrent sets consisting of an attracting closed orbit a, a repelling closed orbit r and a saddle set that is a suspension of a full n-shift and draw some conclusions about the knotting and linking of a and r. For example, we show for all values of n it is possible for a and r to be unknots. For any even value of n it is possible for a and r to be the Hopf link, a trefoil and meridian, or a figure-8 knot and meridian.
Abstract. This is an expository account of a theorem of Louise Moser that describes the types of manifolds that can be constructed via Dehn surgery along a trefoil in the 3-sphere. These include lens spaces, connected sums of two lens spaces, and certain Seifert fibered spaces with three exceptional fibers. Various concepts from the topological theory of three dimensional manifolds are developed as needed.
Abstract. We study simple Smale flows on S3 and other 3-manifolds modeled by the Lorenz template and another template with four bands but that still has cross section a full 2-shift.
The aim of this paper is to show that any two knots can be realized as an attractor and repeller pair for some nonsingular Smale flow on S3 with any linking number. We view this as progress, albeit limited, to the conjecture that all two component links can be realized as an attractor-repeller pair in a nonsingular Smale flow on S3 with just one other basic set of saddle type.
This paper is aimed at undergraduate readers. It explores some connections between vector calculus and linear algebra.
In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse-Smale flows on the 3-sphere have transverse foliations. We then observe that there is a connection to flows arising from certain Hamiltonian systems and from certain contact structures.
We show that for m and n positive, composite closed orbits realized on the Lorenz-like template L(m,n) have two prime factors, each a torus knot; and that composite closed orbits on L(-1,-1) have either two for three prime factors, two of which are torus knots.
After a brief survey of various types of flows (Morse-Smale, Smale, Anosov and partially hyperbolic) we focus on Smale flows on the 3-sphere. However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms.
Templates are branched 2-manifolds with semi-flows used to model ``chaotic''hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.
We show that a positive braid is composite if and only if the
factorization is ``visually obvious'' by placing the braid
An expository account of recent progress on twistwise flow equivalence. There is a new result in the appendix.
Let G be a finite group. We classify G-equivariant flow equivalence of nontrivial irreducible shifts of finite type in terms of (i) elementary equivalence of matrices over the integral group ring ZG and (ii) the conjugacy class in ZG of the group of G-weights of cycles based at a fixed vertex. In the case G=Z/2, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of elementary equivalence over ZG. This involves the algebraic K-theory group K1(ZG).
Suppose that p is a nonsingular (fixed point free) C1 flow on a smooth closed 3-dimensional manifold M with H2(M)=0. Suppose that p has a dense orbit. We show that there exists an open dense set N of M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.
We introduce a way of characterizing the linking of one-dimensional minimal sets in three-dimensional flows and carry out the characterization for some minimal sets within flows modeled by templates, with an emphasis on the linking of Denjoy continua. We also show that any aperiodic minimal subshift of minimal block growth has a suspension which is homeomorphic to a Denjoy continuum.
Abstract: Does battlefield residual depleted uranium pose a significant public health risk? How should activists respond?
We define quantum-type invariants for templates that appear in certain dynamical systems. Such invariants are derived from bialgebras and their quantizations called braided Hopf algebras that are defined by Majid. Diagrammatic relations between projections of templates and the algebraic structures are used to define invariants.
We survey results concerning dynamics of flows on the 3-sphere with special attention to the relationship between dynamical invariants and invariants of geometric topology.
A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, Smale flows in the 3-sphere.
We study knotted periodic orbits which are realized in an attractor of a certain ODE first described by Clark Robinson. These knots can be presented so as to have all positive crossings, but may not be restricted to positive braids.
Knots can be factored uniquely into primes, up to order. We hope that our presentation of this classic result will be accessible to advanced undergraduates.
Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.
We consider one dimensional flows which might arise as a hyperbolic invariant set a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.
We prove that a knot which can be represented by a positive braid with a half twist is prime. This is done by associating to each such braid a smooth branched 2-manifold with boundary and studying its intersection with a would-be cutting sphere.
An analysis of the issue of the postmodern journal Social Text in which physicist Alan Sokal published his spoof article.
Connections between knot theory and Stokes' Theorem as presented to students in a sophomore level vector calculus class.
A zeta function for a map f:M ---> M is a device for counting periodic
orbits. For a flow however, there is not a clear meaning to the period
of a closed orbit. We circumvent this for hyperbolic 3-flows which have
A commentary on calculus reform efforts.
Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.
We study an Anosov flow in the complement of the figure-8 knot in the 3-sphere. In 1983 Joan Birman and Bob Williams conjectured that the knot types of the periodic orbits of this flow could have at most two prime factors. Below, we give a geometric method for constructing knots in this flow with any number of prime factors.
R. F. Williams showed that all knots in the Lorenz template are prime. His proof included the cases where any number of positive twists were added to one of the template's branches. However, Williams does give an example of a composite knot in a template with a single negative twist. We will show that in all the negative cases composite knots do exist, and give a mechanism for producing many examples.