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Diagrammatic methods have transformed the study of topology by making complex entangled structures easier to analyze and classify. They represent these structures as compact visual codes that can be manipulated mathematically to determine whether different descriptions represent the same object and to calculate topological invariants.

However, no complete framework existed for 3-periodic entangled structures, which repeat along all three spatial directions. Researchers therefore lacked a rigorous way to determine when different unit-cell descriptions represented the same infinite structure, limiting the development of a systematic mathematical theory.

“There is no privileged direction along which a unit cell can be projected to obtain a diagram”, explains Sonia Mahmoudi, the lead investigator of an AIMR research team. “Any two diagrams obtained from unit cells must be considered equivalent in a way that reflects not only deformations within the unit cells themselves, but also periodic deformations of the infinite structures the unit cells represent.”

In a 2025 article, Mahmoudi and coworkers introduced a new framework for 3-periodic tangles¹. The researchers developed a new representation called a tridiagram, which captures a 3-periodic tangle as three complementary projections. Together with a complete set of topology-preserving transformations, this representation provides a rigorous way to determine when different descriptions represent the same underlying structure.

Using this approach, the researchers established a generalized Reidemeister theorem for 3-periodic tangles, proving that two tridiagrams represent the same tangle if and only if, up to affine transformations of space such as shearing and scaling, they are related by the prescribed transformations. They also defined a crossing number, extending one of the fundamental invariants of classical knot theory to 3-periodic tangles.

“This gives a new foundation for studying entangled structures, including topological models of crystalline materials, using knot-theoretical methods”, says Mahmoudi. “By giving these structures a rigorous diagrammatic description, we hope to make their topology as accessible to mathematical analysis as classical knots.”

Looking ahead, the team aims to develop computable invariants and complexity measures within the tridiagram framework to compare periodic structures and detect topological equivalence. Longer term, they hope this foundation will help connect topology to structure and function in materials such as crystalline solids, polymers, and DNA-based systems.

A personal insight from Dr. Sonia Mahmoudi

Is there a memory or moment from this project that stands out to you?

One memory that stands out is visiting my colleague Myfanwy Evans in Potsdam, where this project began—not at a whiteboard, but at a café table, with metal cube frames wrapped in wool, trying to brainstorm how to formalize the ideas that became this paper. It’s a memory I still return to: starting from something physical, sitting right there on the table, and slowly working toward the mathematics that would describe it.

(Author: Patrick Han)

This research highlight has been approved by the authors of the original article and all information and data contained within has been provided by said authors.