Decoding doubly periodic weaves: A new era in topological classification

Doubly periodic weaves—entangled structures with repeating patterns in two independent directions—pose a mathematical challenge. Originally conceived to model real-world structures, such as woven textiles and molecular weaving of polymers, mathematicians have generalized the theory to include weaves with any number of distinct directions, extending beyond practical weaving to a broader topological framework.

However, one major obstacle has been the lack of suitable invariants to classify periodic weaves consistently. Traditional knot invariants struggle to capture the complexity of repeating patterns, particularly in distinguishing different weaves or defining minimal units for analysis.

In a 2023 article, Sonia Mahmoudi and co-workers from AIMR introduced crossing matrices and adapted the concept of crossing numbers to periodic structures¹. These tools offered a framework for the topological classification of doubly periodic untwisted weaves, bridging gaps in prior approaches by including both biaxial and triaxial weaves.

“The use of crossing matrices enabled us to encode the arrangement of crossings in periodic weaves systematically and to compute their crossing number within their minimal unit cells,” explains Mahmoudi. “This provided a robust combinatorial method for analyzing and classifying these topological structures.”

The team demonstrated this by calculating the total crossing number of a weave precisely and by establishing equivalence classes of weaves through cyclic permutations of matrix rows and columns.

These results paved the way for advancements—beyond mathematics—in materials science, such as creating more resilient fabrics or designing nanostructures with tailored properties.

Future research aims to extend these methods to more complex periodic entanglements, opening exciting new possibilities in both theoretical and applied sciences, including technical fabrics, molecular weavings, and metamaterials.

This article was written by Patrick Han, Ph.D. (patrick@sayedit.com).

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Advanced Institute for Materials Research (AIMR)
Tohoku University
Establishing a World-Leading Research Center for Materials Science
AIMR aims to contribute to society through its actions as a world-leading research center for materials science and push the boundaries of research frontiers. To this end, the institute gathers excellent researchers in the fields of physics, chemistry, materials science, engineering, and mathematics and provides a world-class research environment.

AIMR site: https://www.wpi-aimr.tohoku.ac.jp/en/