Two passengers step into an empty car on a ring subway line and take their seats. After one full loop, they are surprised to find themselves in each other’s seats — even though nobody moved. That’s pretty much how monodromy plays out in vineyards — families of persistence diagrams, a method rooted in the work of Henri Poincaré. Each point in the diagram is born and dies at its own scale, but when the parameter is wrapped around a loop, feature identifiers can get mixed up. This confusion is especially dangerous in data analysis from biology, climatology, or materials science, where tracking each object through the entire cycle is critical.

To find the root of the problem, mathematicians turned to singularity theory — a toolkit for classifying critical points. They considered a smooth closed curve in the plane and a family of squared distance functions to a point on the loop. The key tool was the symmetry set: the locus of centers of circles tangent to the curve at two points (or having second-order contact). In essence, it’s the skeleton of the shape, whose singularities are fully described by the Bruce–Giblin–Gibson classification. The evolution of the whole system is governed by differential equations on the parameter space. The researchers tracked the information about the birth and death order of topological loops for each of the eight singularity types.

The result turned out to be surprisingly clean. Local monodromy is caused only by the A₂₁/A₂₁ singularity — the intersection of two smooth branches of the symmetry set. One branch corresponds to the simultaneous birth of two connected components, the other to the simultaneous death of two cycles. In the neighborhood of such a point, the space splits into four sectors, like a clock face divided into quarters. Two concentric circles appear, with four points of tangency. As the parameter makes a full loop around the singularity, these four points sequentially shift their order. Then, by the elder rule, two points on the diagram swap places — an exchange occurs. Numerical simulations confirm: intertwined “vineyard vines” yield only such singularities; all others produce unlinked rings.

Now we know where to look for the source of “mix-ups” in cyclic data. The result for the first time links the fine geometry of the original shape — the singularities of its symmetry set — with the topology of the vineyard. This not only explains known examples but also opens the door to algorithmic control: we can deliberately design loops that capture the desired singularities to manage feature tracking, or avoid them to simplify the analysis. A natural next step is extending the method to three-dimensional surfaces and beyond, where singularity classification is already partly complete. There is also a global challenge: even with a local A₂₁/A₂₁ singularity, a large loop might not produce monodromy if the singularities interact “out of phase.” Solving this will require nonlinear dynamics and perhaps knot theory.

Let’s return to the subway car. Now we understand why the passengers swapped seats: the train passed through a special transfer station — a singularity where birth and death lines intersect. Each full loop around it silently swaps the identifiers, and we see two people in each other’s seats. Monodromy in the vineyard is like the map of such a ring line, where the track geometry dictates who ends up where after a loop. Once we’ve figured out the station’s design, we hold the key to designing data routes with no surprises.