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Fabrics are made by repeatedly intertwining yarns into characteristic patterns. Many of their properties, such as stretchiness, arise not only from the material itself but also from how the yarns are arranged and entangled. Such properties illustrate how topology—the underlying patterns of connectivity and entanglement within a structure—can shape a material's overall behavior. Understanding these relationships could help researchers design materials with tailored properties through the design of their topology.
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A research team led by Dr. Daisuke S. Shimamoto, a senior researcher at the Research Organization of Science and Technology, Ritsumeikan University, Japan, along with Dr. Keiko Shimamoto, an independent researcher from Tokyo, Japan, Dr. Sonia Mahmoudi from Tohoku University, and Dr. Samuel Poincloux from Aoyama Gakuin University, has developed a mathematical framework based on knot theory for characterizing knittability and classifying periodic textile structures based on how defects spread through them. Their findings were published in Physical Review X on July 14, 2026.
Tracing how defects travel
The researchers found that defects appear as disruptions in repeating textile patterns and spread through structures in distinct ways depending on their topology. By analyzing these propagation patterns, the framework can determine whether a textile structure is knittable and classify different types of periodic textiles. According to Shimamoto, understanding defect propagation could guide the design of novel fabrics with tailored mechanical properties.
"These defects appear as disruptions in repeating stitch patterns and spread through the structure in distinct ways. This process of propagation of defects described in our framework could guide the design of novel knitted materials with unusual mechanical properties and improve our understanding of other systems shaped by topology," says Shimamoto.
From stitch patterns to knots
The researchers represented knitted and crocheted fabrics as two-dimensional textile diagrams composed of one-dimensional curves, modeling them as repeating, grid-like patterns of interconnected loops. They then introduced defects into the repeating pattern and analyzed how they propagated through neighboring regions of the textile without the yarn being damaged.
To determine whether a textile was knittable, they folded the resulting defect-containing pattern onto a doughnut-shaped surface called a torus and examined whether the resulting knot or link could ultimately be disentangled into simple loops without crossings. A textile was considered knittable if defect propagation transformed the structure into a topologically trivial knot or link.
Using this framework, the researchers were able to distinguish loop-based structures, such as knitting and crochet, from other textile classes. They also found that by controlling how defects spread through a textile, they could influence how damage develops within the structure, providing a new route to fabrics with tunable mechanical properties such as damage resistance.
Designing for damage or durability
Based on these principles, the researchers also designed textiles that suppress defect propagation and therefore undergo limited damage. Similarly, they designed textiles that amplify damage propagation and unravel easily, demonstrating the advantage of mathematically understanding topology.
Beyond textiles, the framework offers a new way to explore how topology influences the behavior and mechanical properties of entangled systems, including polymers, biological tissues and soft robotic materials.
"Our study connects traditional textile crafts with modern mathematics and physics. It provides a systematic way to explore and design textile structures based on topology. It could help develop more durable fabrics without changing the material itself, simply by modifying the entanglement pattern. Since entanglements appear in many systems beyond textiles, including polymers, biological tissues, and soft robotics, it may also inspire new approaches to designing and understanding complex materials," says Shimamoto.
More information: Daisuke S. Shimamoto et al, Topological Defect Propagation to Classify Knitted Fabrics, Physical Review X (2026). DOI: 10.1103/g565-3dyn
Provided by Ritsumeikan University
This story was originally published on Phys.org.
