[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81793-en":3,"doc-seo-81793-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81793,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","The Singular Source of Vineyard Monodromy","Vineyards, or time-varying families of persistence diagrams, enable tracking topological features in topological data analysis as a parameter varies. When the parameter follows a closed loop, diagram points may undergo monodromy, permuting over one traversal and hindering feature tracking. This work studies monodromy for 1-manifolds in R2 via singularity theory and gives a classification: a sufficiently small loop cannot induce monodromy unless it contains a specific singularity of the distance function determined by the symmetry set.","arXiv :2607 .0 1046v 1 [ cs .CG] 1 Jul 2026  \n1 The Singular Source of Vineyard Monodromy  \n2 Erin W. Chambers∗ Christopher Fillmore† Shankha Shubhra Mukherjee ‡  \n3 Rohit Roy § Elizabeth Stephenson¶ Mathijs Wintraecken ‖  \n4 July 2, 2026  \n5 Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA)  \n6 pipelines to track how topological features change and evolve as a parameter varies. When the parameter traces a closed  \n7 loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature  \n8 tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise  \n9 from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram  \n10 fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy  \n11 and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in  \n12 two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why  \n13 monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines. In  \n14 this work, we study this question for 1-manifolds in R2 , using a surprising connection with tools from singularity theory, 15 and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a  \n16 sufficiently small loop γ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The  \n17 central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than  \n18 one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely  \n19 when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting  \n20 and utilizing (or avoiding) monodromy in TDA pipelines.  \n21 Keywords: Symmetry set, persistent homology, vineyards  \n22 1 Introduction Topological Data Analysis (TDA) encompasses a wide range of tools that compute  \n23 topological invariants of geometric objects or spaces using tools from algebraic topology. Persistent homology in  \n24 particular is perhaps the most well-known example in TDA, yielding a computable and stable invariant known  \n25 as the persistence diagram, which has seen wide utility in data analysis in a broad range of applications. A full  \n26 review of such applications is beyond our scope, but we refer the reader to [42] for a list of further examples; see  \n27 also recent books on this topic [12, 29] .  \n28 Time-series data yields a family of stacked persistence diagrams known as vineyards [24, 61], which have been  \n29 useful in many applications [11, 28, 64] . Because of the stability of persistence diagrams [22], the points in the  \n30 stacked persistence diagrams move continuously over time. This means that we can follow a point in (the stack 31 of) the persistence diagrams; the resulting curve is called a vine. Vineyards can be computed via a modified  \n32 version of the standard persistence algorithm [24] . Here we will call the family or ‘circular stack’ of persistence  \n33 diagrams of a continuous periodic family of filtrations a closed vineyard [16] .  \n∗ Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA ([echambe2@nd.edu](echambe2@nd.edu), [https://wolfchambers.github.io/](https://wolfchambers.github.io/) , [https://orcid.org/0000-0001-8333-3676](https://orcid.org/0000-0001-8333-3676)).  \n†Institute of Science and Technology Austria, Klosterneuburg, Austria (cdfillmore@gmail.com, https://orcid.org/ 0000-0001-7631-2885) .  \n‡Department of Computer Science and Engineering, Uni","cbCaiqILHMsI21Hq","https://ap.wps.com/l/cbCaiqILHMsI21Hq","pdf",7457705,1,28,"English","en",105,"# Introduction\n## Vineyards and closed vineyards\n## Monodromy and prior work","[{\"question\":\"What is vineyard monodromy in topological data analysis?\",\"answer\":\"In closed vineyards, points in persistence diagrams can permute after one full loop of the parameter. This monodromy obstructs consistent feature tracking across the traversal.\"},{\"question\":\"Which geometric setting and loops are analyzed in this work?\",\"answer\":\"The study focuses on 1-manifolds in R2 and considers sufficiently small loops γ. The results characterize when such loops can or cannot produce monodromy.\"},{\"question\":\"What is the key object used to determine monodromy occurrences?\",\"answer\":\"The symmetry set is the central geometric object. It classifies singularities of the distance function and dictates precisely when monodromy occurs.\"}]",1784176184,71,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"the-singular-source-of-vineyard-monodromy","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/the-singular-source-of-vineyard-monodromy/81793/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is vineyard monodromy in topological data analysis?","Question",{"text":74,"@type":75},"In closed vineyards, points in persistence diagrams can permute after one full loop of the parameter. This monodromy obstructs consistent feature tracking across the traversal.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Which geometric setting and loops are analyzed in this work?",{"text":79,"@type":75},"The study focuses on 1-manifolds in R2 and considers sufficiently small loops γ. The results characterize when such loops can or cannot produce monodromy.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the key object used to determine monodromy occurrences?",{"text":83,"@type":75},"The symmetry set is the central geometric object. 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