Counts and end-curves in two-parameter persistence

2025-05-19

AI Summary

This paper addresses the absence of a structured invariant in two-parameter persistence theory analogous to the “barcode” in single-parameter persistence. Method: It introduces the *count* and *end-curves*—novel invariants for two-graded modules—by integrating two-graded module theory, Betti table interpolation, representations of string algebras, inclusion–exclusion principles, and multiparameter homological analysis over the ring of real-exponential polynomials. Contributions: (1) It proves the count is uniquely determined and equals the number of end-curves; (2) it establishes that end-curves interpolate generators, relations, and syzygies; and (3) it proposes the *boundary*, a complete invariant for spread-decomposable representations, which is incomparable with the rank invariant—thereby filling a fundamental gap in the structural characterization of two-parameter persistence modules.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of bar-count and endpoints of singly-graded modules over the polynomial ring in one variable, from persistence theory. We show that our count is the unique one satisfying certain natural conditions; as a consequence, several inclusion-exclusion-type formulas in two-parameter persistence yield the same positive number, which equals our count, and which in turn equals the number of end-curves, giving geometric meaning to this count. We show that the end-curves determine the classical Betti tables by showing that they interpolate between generators, relations, and syzygies. Using the band representations of a certain string algebra, we show that the set of end-curves admits a canonical partition, where each part forms a closed curve on the plane; we call this the boundary of the module. As an invariant, the boundary is neither weaker nor stronger than the rank invariant, but, in contrast to the rank invariant, it is a complete invariant on the set of spread-decomposable representations. Our results connect several lines of work in multiparameter persistence, and their extension to modules over the real-exponent polynomial ring in two variables relates to two-dimensional Morse theory.

Problem

• Defining two-parameter count and end-curves for bigraded modules
• Linking end-curves to classical Betti tables and syzygies
• Establishing boundary as complete invariant for spread-decomposable representations

Innovation

• Defines two-parameter count and end-curves
• Links end-curves to classical Betti tables
• Introduces boundary as complete invariant