Periodic prime knots and topologically transitive flows on 3-manifolds
type: Item
instance of: scholarly article
title: Periodic prime knots and topologically transitive flows on 3-manifolds
zbMATH Open document ID: 1098.37019
author: Michael C. Sullivan
author: William F. Basener
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: Given a smooth nonsingular topologically transitive flow on a closed 3-manifold with integer homology zero in dimension 2, then by the main result there is an open dense subset such that any closed orbit intersecting the subset is a nontrivial prime knot. Here, in transferring notions from classical knot theory, a knot is trivial if it bounds a disk, a knot is composite if it meets a 2-sphere in two points that reduce the knot to two nontrivial ones, and a knot is prime if it is nontrivial but not composite. The above result is deduced from work of the first author [Topology Appl. 135, 131--148 (2004; Zbl 1032.37007), and Topology 43, 697--703 (2004; Zbl 1079.57002)] as well as a technical method involving foliations by \textit{C. Gutierrez} [Topology 34, 679--698 (1995; Zbl 0842.57032)].
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 5013536
zbMATH Keywords: dense orbit
zbMATH Keywords: minimal flow
zbMATH Keywords: global cross section
zbMATH Keywords: prime knots
zbMATH Keywords: closed orbit
zbMATH Keywords: foliation
zbMATH Keywords: global transverse disk
zbMATH Keywords: topologically transitive flow
Wikidata QID: Q57441623
reviewed by: Dieter Erle
MaRDI profile type: Publication
full work available at URL: j.topol.2005.03.009
OpenAlex ID: W2092790690
cites work: Every nonsingular \(C^{1}\) flow on a closed manifold of dimension greater than two has a global transverse disk
cites work: Knots and topologically transitive flows on 3-manifolds.
cites work: Knots and links in three-dimensional flows
cites work: Knots and minimal flows on 3-manifolds
cites work: A fixed-point free ergodic flow on the 3-sphere
DOI: 10.1016/J.TOPOL.2005.03.009
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description: scientific article; zbMATH DE number 5013536