Fat handles and phase portraits of Non Singular Morse-Smale flows on S^3 with unknotted saddle orbits

TL;DR

This paper constructs non-singular Morse-Smale flows on the 3-sphere with unknotted saddle orbits using fat handle decompositions, revealing how heteroclinic trajectories impose an order on their phase portraits.

Contribution

It introduces a method to build and analyze Morse-Smale flows on S^3 with unknotted saddle orbits via fat handle identification, elucidating the role of heteroclinic trajectories.

Findings

Heteroclinic trajectories impose an order on handle decompositions.

The order is total for flows with one repulsive, one attractive, and multiple unknotted saddle orbits.

The construction enables explicit phase portrait analysis.

Abstract

In this paper we build Non-singular Morse-Smale flows on S^3 with unknotted and unlinked saddle orbits by identifying fat round handles along their boundaries. This way of building the flows enables to get their phase portraits. We also show that the presence of heteroclinic trajectories imposes an order in the round handle decomposition of these flows; this order is total for NMS flows composed of one repulsive, one attractive and n unknotted saddle orbits, for n >1.