Encoding and Computation on Textiles

We model a textile as an infinite collection of lines and circles whose embedding in a thickened plane maps to itself under periodic translation in two linearly independent directions. Selection of a fixed unit cell allows a finite representation of a textile in a fixed thickened torus, from which we derive an abstract encoding of its structure based on the Gauss Code for knots.
This enables an algorithmic approach to enumerating viable real world textile structures, and we will present our application of this approach to enumerating all oriented textiles with a given number of crossings, which is our main result to date on textile structures.
However, unit cell selection in periodic structures is not unique, and topological classification should therefore be up to both deformation of the ambient space and changes in unit cell selection – this notion of periodic isotopy will be the key tool used to study real world examples of periodic structures. We will discuss our preliminary approach to this topic, which has not to date received much study.