Braiding vineyards

In this work, we introduce and study what we believe is an intriguing and, to the best of our knowledge, previously unknown connection between two areas in computational topology, topological data analysis (TDA) and knot theory. Given a function from a topological space to $\mathbb{R}$, TDA provides tools to simplify and study the importance of topological features: in particular, the $l^{th}$-dimensional persistence diagram encodes the $l$-homology in the sublevel set as the function value increases as a set of points in the plane. Given a continuous one-parameter family of such functions, we can combine the persistence diagrams into an object known as a vineyard, which track the evolution of points in the persistence diagram. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the study of monodromy, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. In this work, given a link and value $l$, we construct a topological space and periodic family of functions such that the closed $l$-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope. Importantly, it has at least two immediate consequences: First, monodromy of any periodicity can occur in a $l$-vineyard, answering a variant of a question by [Arya et al 2024]. To exhibit this, we also reformulate monodromy in a more geometric way, which may be of interest in itself. Second, distinguishing vineyards is likely to be difficult given the known difficulty of knot and link recognition, which have strong connections to many NP-hard problems.