Encoding and Topological Computation on Textiles

TL;DR

This paper introduces a new encoding method for textile structures, enabling efficient topological classification and analysis, including a linear time algorithm for physical realizability and a comprehensive classification up to complexity five.

Contribution

It extends the Gauss code to textile structures, providing a novel combinatorial encoding and classification tools for periodic woven textiles.

Findings

Developed a linear time algorithm for textile physicality check.

Classified all oriented single-component textiles up to complexity five.

Extended classical knot invariants to textile structures.

Abstract

A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The first important problem is to encode all textile structures in a simple combinatorial way. This paper extends the notion of the Gauss code in classical knot theory, providing a tool for topological computation on these structures. As a first application, we present a linear time algorithm for determining whether a code represents a textile in the physical sense. This algorithm, along with invariants of textile structures, allowed us for the first time to classify all oriented textile structures woven from a single component up to complexity five.