Simple Smale flows and their templates on $S^3$

TL;DR

This paper establishes a complete isotopic invariant for embedded templates in $S^3$ using spatial graph invariants, aiding the classification of simple Smale flows on the 3-sphere.

Contribution

It introduces a new invariant based on Kauffman's spatial graph invariant for embedded templates, advancing the classification of Smale flows.

Findings

Boundary of embedded templates is a complete isotopic invariant.

Constructs an invariant of templates using Kauffman's spatial graph invariant.

Classifies simple Smale flows on $S^3$ using the developed invariants.

Abstract

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an embedded template in $S^3$ with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set of knots and links. As application, the isotopic classification of simple Smale flows on $S^3$ is discussed.