Discrete analog of the Jacobi set for vector fields

TL;DR

This paper introduces a discrete, piecewise linear version of the Jacobi set for vector fields on simplicial complexes, extending previous concepts from scalar functions to vector fields.

Contribution

It generalizes the Jacobi set concept from scalar functions to vector fields on simplicial complexes using a piecewise linear approach.

Findings

Defines the piecewise linear Jacobi set for vector fields

Extends the concept from gradients of functions to general vector fields

Provides a framework for analyzing vector fields on simplicial complexes

Abstract

The Jacobi set is a useful descriptor of mutual behavior of functions defined on a common domain. We introduce the piecewise linear Jacobi set for general vector fields on simplicial complexes. This definition generalizes the definition of the Jacobi set for gradients of functions introduced by Edelsbrunner and Harer.