Twisted topological tangles or: the knot theory of knitting
May 23, 2023
Event: Tangled in Knot Theory
Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure. At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity. The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. We have developed a formalization of the topology of two-periodic weft knitted textiles using a construction we call the swatch . Using this construction, we can prove that all two-periodic weft knits form ribbon links . This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics. The authors were partially supported by National Science Foundation grant DMR-1847172, by the Research Corporation for Science Advancement and by the International Center for Sustainability with Chiral Knotted Meta Matter (SKCM²). We would like to thank sarah-marie belcastro, Jen Hom, Jim McCann, Agniva Roy, Saul Schleimer and Henry Segerman for many fruitful conversations.  S. Markande and S. Matsumoto, in: Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Culture, (Tesselations Publishing, 2020), pp. 103–112.  M. Kuzbary, S. Markande, S. Matsumoto and S. Pritchard, 2022.