Approaching knitting through "topology": Ritsumeikan University and partners devise fabric resistant to running

A research group at Ritsumeikan University and other institutions has clarified a method for determining, from the perspective of topology, whether textiles with periodically entangled yarns can be produced as knitted fabrics. They also found that the fragility by which a hole spreads outward from an unraveled section of yarn is closely tied to the fundamental nature of knitting. The findings could enable the systematic discovery of new textiles, including knitting techniques for run-resistant stockings.

In knitting, fabric is created by repeatedly looping and interlocking a single strand of yarn using knitting needles. About two years ago, Senior Researcher Daisuke Shimamoto (Physics) of the Research Organization of Science and Technology at Ritsumeikan University began investigating whether knitting could be defined from the perspective of topology. Topology is a branch of mathematics that deals with properties preserved under deformation without cutting or joining.

From a topological perspective, a structure can be regarded as knitting if it can be created while keeping both ends of the yarn fixed. In contrast, woven fabrics, in which weft yarns pass through warp yarns, as well as fences and nets made by twisting or knotting wire, are not considered knitting.

For established knitting methods such as the widely used stockinette stitch, it can be shown that they can be created while keeping both ends of the yarn fixed. However, for fabrics with previously unknown structures, there had been no general method for determining whether they could be produced as knitted fabrics. Shimamoto turned his attention to knot theory, a branch of topology. Knot theory classifies entangled string-like curves in space according to whether they can be deformed without being cut.

Based on knot theory, the researchers treated the periodic entanglements found in fabrics as repetitions of small cells and examined what happens to the surrounding cells when a single cell is unraveled. They found that knitted fabrics share a common feature: a change occurring at one location propagates in a chain reaction to neighboring regions. For example, fabrics that unravel from the point where a yarn breaks, as in a run in a stocking, are classified as knitting.

Shimamoto said, "In defining knitting from the perspective of topology, we found that the essence of knitting lies in its fragility, which is inseparable from its ease of construction." To eliminate fragility while preserving ease of construction, he devised a structure in which "for a given cell to unravel, multiple surrounding cells must unravel simultaneously." As a result, even when part of the fabric unraveled, the damage did not continue indefinitely in the vertical direction and remained confined to a limited-sized hole.

The study, conducted in collaboration with researchers from Tohoku University, Aoyama Gakuin University, and other institutions, was published in the international journal Physical Review X on July 14.