Visualization of Morse flow with two saddles on 3-sphere diagrams

TL;DR

This paper classifies all topological structures of Morse-Smale flows on a 3-sphere with specific saddle and connection configurations, using generalized diagrams and visualizations.

Contribution

It provides a complete classification of Morse-Smale flows with two saddles and specific connection constraints on a 3-sphere, employing generalized Heegaard diagrams.

Findings

All possible embeddings of two circles with up to 10 intersection points identified.

Constructed planar visualizations of these embeddings.

Classified topological structures up to homeomorphism.

Abstract

We describe all possible topological structures of Morse-Smale flows without closed trajectories on a three-dimensional sphere, which have two sources, two sinks, one saddle of Morse index 1, one saddle of Morse index 2, and no more than 10 saddle connections. To classify such flows, a generalized Heegaard diagram or Pr-diagram is used, which in this case consists of a sphere and two closed curves, the intersection points of which correspond to saddle connections. We have found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection and construct its planar visualisations.