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

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.