The universal templates of Ghrist. (English) Zbl 0902.57001

Williams, R. F.

Bull. Am. Math. Soc., New Ser. 35, No. 2, 145-156 (1998).

The author reports on some recent work of Robert W. Ghrist [Topology 36, No. 2, 423-448 (1997; Zbl 0869.57007)] in which Ghrist shows that there are many structurally stable flows on the 3-sphere, each of which has closed orbits representing every knot type. This answers a question originally asked by M. Hirsch. In fact, Ghrist proves a stronger result, namely he constructs flows on the 3-sphere which contain all closed braids as unions of periodic orbits. The work of Ghrist makes use of the notion of template for a flow, which was first introduced in a paper of J. S. Birman and R. F. Williams [Topology 22, 47-82 (1983; Zbl 0507.58038)]. The notion of template itself originates in a paper of R. F. Williams [Colloque Topol. différ., Univ. Montpellier, Publ. 55 (1968-1969), 79-89 (1969; Zbl 0208.25801)], a template being a branched 2-manifold with boundary, lying in the 3-sphere, which is endowed with a smooth vector field. The result of Ghrist on which the author reports can be stated as follows: There exist universal templates. In this report, the author describes some background material, and makes a sketch of the history of the problem. He gives then an account of the main ideas in the work of Ghrist. The paper ends with a list of unsolved problems related to the results described.

Reviewer: A.Papadopoulos (Strasbourg)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
37C10	Dynamics induced by flows and semiflows

Keywords:

link; knot; template; universal template; branched 2-manifold; periodic orbit; braid; flow

Citations:

Zbl 0869.57007; Zbl 0507.58038; Zbl 0208.25801

References:

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