Chain Mail, Knitting, and One- and Two-Periodic Knots

Date started: September 2019
Leads: Frank Farris and Max Krause


During the Computational Textiles workshop at ICERM, Farris realized that Fourier series techniques for creating symmetry (see Creating Symmetry reference below) could be applied to construct virtual symmetric knitting patterns and even chain mail, using Grasshopper and Rhino.  Seeing Farris’s early creations, Krause realized that these could provide examples for his ongoing project on one- and two-periodic knots, connecting to previous work by others (see Champanerkar reference below). Krause’s mathematical analysis can lead Farris to create patterns with interesting knot properties; Farris’s construction techniques can help Krause create his own examples. One of the central mathematical questions is: Can the symmetry of a chain mail (or knitted or woven) knot be detected from knot or link invariants?


3D printed chain mail with symmetry group cm (also called *x).
Virtual chain mail with symmetry group cmm (also called 2*22).
Virtual demonstration of knitting in “ribbing” stich.
Krause’s virtual construction of a 2-periodic knot with the “Borromean rings” property: cutting any one string will cause the whole pattern to become unknotted.


  • A. Champanerkar, I. Kofman, and J. Purcell, “Geometry of biperiodic alternating links.” J. Lond. Math. Soc., 99:3, June, 2019, pp. 807–830.
  • F. Farris. Creating Symmetry: The Artful Mathematics of Wallpaper Patterns.  Princeton, 2015.