Given a manifold M without boundary, the Radial Persistence Transform (RPT) assigns to each point x in R^2 the persistence diagram of the squared Euclidean distance function d_E^2(., x) restricted to M. As x moves along a closed loop gamma, these diagrams form a periodic vineyard. The RPT is a variant of the persistent homology transform (PHT) introduced by Turner and collaborators, and is more sensitive, see Onus et al. Both the RPT and PHT can exhibit monodromy; after a full traversal of gamma, points of the persistence diagram return permuted. Monodromy obstructs feature tracking and complicates downstream analysis in TDA pipelines, and understanding when and why it occurs is an open problem. Arya et al. gave an example with monodromy for the PHT and the geometric conditions that exclude monodromy, both in 2 dimensions.

Chambers et al. showed that monodromy and knotting of arbitrary complexity can occur for the RPT. We are exploring the relation between the monodromy in a vineyard and the topology of the vineyard for the RPT on one hand and the symmetry and focal set (the bifurcation set for distances) on the other. Classical results of Bruce, Giblin and Gibson, and of Porteous, show that the Morse decomposition of the distance function changes only when x crosses the symmetry set or the focal set of M. We prove that for curves in R^2, the vineyard of a sufficiently small loop gamma cannot exhibit monodromy unless gamma encloses a specific singularity of the symmetry set. This yields a local classification of the causes of monodromy in 2 dimensions, which we are currently expanding to a global classification. This opens the door to algorithmic criteria for detecting and then exploiting or avoiding it.

Our investigation has been informed and corroborated by experimentation with Bouquet2D, an interactive web-based tool for exploring this phenomenon. The tool allows the user to define M as a spline curve and draw a loop gamma, and the tool displays the symmetry and focal sets of M together with the vineyard as x moves along gamma. Deforming M updates all of these in real time, so one can directly observe how bifurcations of the symmetry and focal sets appear as events in the vineyard, and where monodromy arises. Bouquet2D is available for everyone to experiment with vineyard topology at: https://bouquet2d.rohitroy.me/.