Notes on Pointwise Finite-Dimensional $2$-Parameter Persistence Modules

TL;DR

This paper investigates finite-dimensional 2-parameter persistence modules, demonstrating their decomposition into constant functors over chambers, finite encodings, and characterizing indecomposable thin modules as polytope modules.

Contribution

It introduces a decomposition framework for p.f.d. 2-parameter modules using convex isotopy subdivisions and characterizes indecomposable thin modules as polytope modules.

Findings

Decomposition of modules into constant functors over chambers

Finite encoding of modules via convex isotopy subdivisions

Indecomposable thin modules are polytope modules

Abstract

In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module $M$ (with a finite convex isotopy subdivision) is isomorphic to a $2$-parameter persistence module $N$ where the restriction of $N$ to each chamber of the parameter space $(\mathbb{R},\leq)^2$ is a constant functor. Moreover, we show that the convex isotopy subdivision of $M$ induces a finite encoding of $M$. Finally, we prove that every indecomposable thin $2$-parameter persistence module is isomorphic to a polytope module.