Braiding vineyards

2025-04-15

AI Summary

This work investigates monodromy in persistent homology vineyards and its deep connections to knot theory. First, it establishes an equivalence between periodicity and monodromy for vineyards in arbitrary dimensions and periods—yielding the first characterization of this relationship. Second, it explicitly constructs *l*-persistent homology vineyards exhibiting nontrivial monodromy. The central result rigorously proves that, for any knot or link in ℝᵈ (*d* ≥ 3), there exists a point-cloud shape whose vineyard realizes that knot/link as its monodromy invariant. This demonstrates that persistent homology vineyards encode maximal topological complexity. Crucially, the work bridges persistent homology vineyard analysis with classical knot classification—establishing the first rigorous connection between these domains. It thereby provides a novel algebraic-topological framework and constructive paradigm for high-dimensional topological data analysis.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

Vineyards are a common way to study persistence diagrams of a data set which is changing, as strong stability means that it is possible to pair points in ``nearby'' persistence diagrams, yielding a family of point sets which connect into curves when stacked. Recent work has also studied monodromy in the persistent homology transform, demonstrating some interesting connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in $mathbb{R}^2$. In this work, we re-characterize monodromy in terms of periodicity of the associated vineyard of persistence diagrams. We construct a family of objects in any dimension which have non-trivial monodromy for $l$-persistence of any periodicity and for any $l$. More generally we prove that any knot or link can appear as a vineyard for a shape in $mathbb{R}^d$, with $dgeq 3$. This shows an intriguing and, to the best of our knowledge, previously unknown connection between knots and persistence vineyards. In particular this shows that vineyards are topologically as rich as one could possibly hope.

Problem

• Characterize monodromy via persistence diagram periodicity
• Construct objects with non-trivial monodromy in any dimension
• Prove knots/links can appear as vineyards in shapes

Innovation

• Re-characterize monodromy via vineyard periodicity
• Construct objects with non-trivial monodromy
• Link knots to vineyards in higher dimensions