Non-singular flows with twisted saddle orbit on orientable 3-manifolds

TL;DR

This paper classifies non-singular Morse-Smale flows with a single twisted saddle orbit on orientable 3-manifolds, identifying the types of manifolds that admit such flows and counting their equivalence classes.

Contribution

It provides a complete topological classification of these flows and determines the number of equivalence classes on each admissible manifold.

Findings

Manifolds are either lens spaces, connected sums of lens spaces with projective spaces, or Seifert manifolds with base sphere.

A full classification of the flows is achieved.

The number of equivalence classes on each manifold is calculated.

Abstract

In this paper we consider non-singular Morse-Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold admitting such flows is either a lens space, or a connected sum of a lens space with a projective space, or Seifert manifolds with base sphere and three special layers. A complete topological classification of the described flows is obtained and the number of their equivalence classes on each admissible manifold is calculated.