Structural stability and generic transitions of "incompressible" line fields on surfaces

TL;DR

This paper introduces a topological framework for analyzing the stability and generic transitions of incompressible line fields on surfaces, with applications to physical and biological systems like active nematics.

Contribution

It develops a new topological approach and invariants for generic line fields, enabling combinatorial analysis of their transitions and stability on surfaces.

Findings

Line fields with certain singularities are shown to be generic under specific conditions.

A new representation of invariants allows encoding line field evolution as walks on transition graphs.

The framework applies to physical systems such as active nematics, demonstrating practical relevance.

Abstract

Various line fields naturally arise on surfaces in both physical and biological contexts, and generic singularities frequently appear in the form of 1-prong (thorn-like) and 3-prong (tripod-like) configurations, which can be modeled by partial differential equations with specific parameter values. However, it remains open under which topologies such line fields are structurally stable and form an open dense subset. In this paper, we propose a new topological framework for describing line fields and their evaluations on surfaces that is suitable from both theoretical and applied perspectives. Specifically, we demonstrate that, under a topology defined by a ``cone'' structure, line fields with 1-prong and 3-prong singularities are generic when an ``incompressibility condition'' holds. We also introduce representations of complete invariants for generic line fields and their generic transitions. These representations enable the evolution of ``incompressible'' line fields -- such as those observed in active nematics -- to be encoded as walks on transition graphs, providing a combinatorial framework for their analysis.