Smale-Fomenko diagrams and rough topological invariants of the Kowalevski-Yehia case

TL;DR

This paper provides a comprehensive classification of the topological invariants and Smale-Fomenko diagrams for the Kowalevski-Yehia gyrostat, identifying all critical point types and their associated graph structures.

Contribution

It introduces a complete analytical classification of atoms at critical points and constructs a method to describe the rough topology of the integrable case.

Findings

Identified all separating values of gyrostatic momentum.

Classified nine groups of identical molecules with stable and unstable graph types.

Provided a complete description of the rough topology of the Kowalevski-Yehia case.

Abstract

We present the complete analytical classification of the atoms arising at the critical points of rank 1 of the Kowalevski-Yehia gyrostat. To classify the Smale-Fomenko diagrams, all separating values of the gyrostatic momentum are found. We present a kind of constructor of the Fomenko graphs; its application gives the complete description of the rough topology of this integrable case. It is proved that there exists exactly nine groups of identical molecules (not considering the marks). These groups contain 22 stable types of graphs and 6 unstable ones with respect to the number of critical circles on the critical levels.