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.

翻译：本文提出并研究了我们认为在计算拓扑学两个领域——拓扑数据分析（TDA）与纽结理论——之间存在的一种引人入胜且据我们所知先前未知的联系。给定从拓扑空间到 $\mathbb{R}$ 的函数，TDA 提供了简化与研究拓扑特征重要性的工具：特别地，$l$ 维持续图将函数值增加时子水平集中 $l$ 维同调编码为平面中的点集。给定此类函数的连续单参数族，我们可以将持续图组合成一个称为藤蔓的对象，它追踪持续图中点的演化过程。若进一步限制该函数族为周期函数，我们可将藤蔓的两端等同起来，从而得到闭合藤蔓。这使得我们可以研究单值性，在此语境下意味着沿函数族运行一个周期会以非平凡方式置换点集。本文中，给定一个链环与值 $l$，我们构造了一个拓扑空间及周期函数族，使得闭合 $l$ 维藤蔓包含该链环。这表明藤蔓在拓扑意义上具有可能达到的最丰富结构。重要的是，这至少带来两个直接推论：首先，任意周期性的单值性均可能出现在 $l$ 维藤蔓中，这回答了 [Arya 等人 2024] 所提问题的一个变体。为展示这一点，我们还以更几何化的方式重构了单值性概念，其本身可能具有独立意义。其次，鉴于纽结与链环识别（其与众多 NP 难问题存在紧密联系）的已知难度，区分不同藤蔓很可能同样困难。