Topological Defect Propagation to Classify Knitted Fabrics

The Wayback Machine - http://web.archive.org/web/20260716053734/https://journals.aps.org/prx/abstract/10.1103/g565-3dynFeatured in PhysicsOpen AccessDaisuke S. Shimamoto*Keiko ShimamotoSonia MahmoudiSamuel PoinclouxGraduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8902, Japan and Research Organization of Science and Technology, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, JapanIndependent researcher, Tokyo, JapanAdvanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan and RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama, 351-0198, JapanDepartment of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan*Contact author: dshima@fc.ritsumei.ac.jpAbstractKnits and crochets are mechanical metamaterials with a long history that can typically be produced from a single yarn. Despite the simplicity of the manufacturing process, they exhibit a wide range of structural configurations with diverse mechanical properties and application potentials. Although there has been recent growing interest in textile-based metamaterials, a rigorous topological characterization of what makes a structure knittable has been lacking. Here, we introduce a general criterion based on topological constraints that distinguish knits and crochets from other textile structures. We demonstrate how the introduction of topological defects and their propagation makes this classification practical. Our approach highlights a fundamental link between manufacturing processes and structural fragility. Within this framework, we show how the rationalization of defect propagation unlocks the design of fabrics with controllable damage resistance.Mechanical metamaterialsTopological materialsTopology synopsisPopular SummaryIn knot theory, the mathematical study of tangled curves, the topology of a knot describes the arrangement of crossings that cannot be undone without cutting the curves.

Fabricating a textile involves entangling yarns into a specific pattern, effectively forming a large knot. We develop a method to determine if a fabric can be knitted based on the topology of its pattern. Knits are produced through local yarn manipulation, keeping the ends fixed. We establish a direct connection between manufacturing constraints and the topological complexity of a knot formed from the textile pattern with some defects. Defects are local yarn disentanglements that propagate in knits only, mimicking in reverse the local yarn manipulations of their fabrication process. Defect propagation disentangles the textile pattern, and it yields only simple knots. In contrast, unknittable patterns yield knots whose complexity scales with their size. Our framework lays a foundation for exploring how topology impacts other textile properties and can help design fabrics with tunable robustness.Article TextReferences (83)J. E. Avron, R. Seiler, and B. Simon, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51, 51 (1983). M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Ann. Phys. (N.Y.) 160, 343 (1985). C. L. Kane and E. J. Mele, Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005). Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018). C. F. Barenghi, Knots and unknots in superfluid turbulence, Milan J. Math. 75, 177 (2007). N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979). R. D. Kamien, The geometry of soft materials: A primer, Rev. Mod. Phys. 74, 953 (2002).

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