arXiv2026
Time-series of persistence diagrams, more commonly known as vineyards, are a useful way to capture how multi-scale topological features vary over time. However, we do lose significant information in how the persistent homology at different time stamps relate to each other. A natural algebraic version of vineyards is a time series of persistence modules equipped with interleaving maps between the persistence modules at different time stamps. We call this a vineyard module. In this paper we will set up the framework for representing a vineyard module via an indexed set of vines alongside a collection of matrices. We outline an algorithmic way to transform the bases of the persistence modules at each time step within the vineyard module to make the matrices within this representation as simple as possible. With some reasonable assumptions on the vineyard modules, this simplified representation can be completely described (up to isomorphism) by the associated vineyard and a vector of finite length. While this vineyard and vector representation is not necessarily unique we can prove that the vector will always be the zero vector when the vineyard module is isomorphic to the direct sum of vine modules. We also show uniqueness when the vineyard module contains two vines (over field $\mathbb{F}_2$) and can characterise and count which are realisable, giving a complete account of isomorphism classes of vineyard modules with two vines. To define and analyse this approach we first must set up a lot of preliminary results about changes of bases for pairs of persistence modules where we are given $\varepsilon$-interleaving maps for sufficiently close $\varepsilon$. This new perspective on vineyards also provides a new approach to the study of multi-parameter persistence modules.