Analysis of Hopf solitons as generalized fold maps

TL;DR

This paper introduces a new mathematical framework using singularity theory and fold maps to model and classify high-Hopf-index hopfions, linking topology with experimental observations.

Contribution

It proposes a generalized Hopf map structure via fold maps and Stein factorizations, providing a rigorous classification of high-Hopf-index hopfions.

Findings

Aligns theoretical models with experimental high-Hopf-index hopfions

Provides a classification scheme for fiber configurations in hopfions

Establishes a connection between singularity theory and physical topological structures

Abstract

The Hopf index, a topological invariant that quantifies the linking of preimage fibers, is fundamental to the structure and stability of hopfions. In this work, we propose a new mathematical framework for modeling hopfions with high Hopf index, drawing on the language of singularity theory and the topology of differentiable maps. At the core of our approach is the notion of a generalized Hopf map of order $n$, whose structure is captured via fold maps and their Stein factorizations. We demonstrate that this theoretical construction not only aligns closely with recent experimental observations of high-Hopf-index hopfions, but also offers a precise classification of the possible configurations of fiber pairs associated to distinct points. Our results thus establish a robust bridge between the geometry of singular maps and the experimentally observed topology of complex field configurations of hopfions in materials and other physical systems.