A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

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This paper proposes a new systematic approach for the description and classification of textile structures based on topological principles. It is shown that textile structures can be considered as a specific case of knots or links and can be represented by diagrams on a torus. This enables modern methods of knot theory to be applied to the study of the topology of textiles. The basics of knot theory are briefly introduced. Some specific matters relating to the application of these methods to textiles are discussed, including enumeration of textile structures and topological invariants of doubly-periodic structures.

This paper is the second in the series on topological classification of textile structures. The classification problem can be resolved with the aid of invariants used in knot theory for classification of knots and links. Various numerical and polynomial invariants are considered in application to textile structures. A new Kauffman-type polynomial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some simplest doubly-periodic interlaced structures and for some woven and knitted textiles.

Yarn structure is one of the important characteristics which define yarn mechanical properties and appearance. For example, it is known that the mechanical properties of a ring-spun and an open-end spun yarn of the same linear density and produced from the same fibers are different [1,2]. This is caused by the difference in their internal structure, which can be characterized by the cross-sectional and longitudinal distribution of fibers.A staple fiber yarn is composed of short-length fibers which are assembled and twisted together. Fibers start at different distances from the beginning of the yarn. The radial positions of fibers within the yarn body change randomly (with unknown probability distribution) along the yarn axis. 1 This behavior is known as fiber migration. Fiber migration is now a universally recognized phenomenon of yarn structure which provides necessary entanglement and cohesion between the fibers, thus creating a continuous structure devised from a discrete set of fibers. This dual nature of the structure of a staple fiber yarn makes it difficult to be modeled. Theoretical and experimental investigations of structure-properties relationship of yarns conducted over Abstract A queuing model of staple fiber yarn is presented that enables the modeling and a better understanding of fiber migration in a yarn. The model provides a fine yarn structure where the migrational behavior of fibers is associated with the behavior of customers traveling across an open network of queuing systems to get services. Based on this analogy, the underlying mathematical foundation of the queuing theory is used for the modeling of yarn structure and properties. The model uses yarn technical specifications including yarn linear density and twist level, fiber linear density and length distribution, together with specific parameters such as fiber packing density distribution and migration probabilities. The model can be used for modeling a wide range of structurally different yarns; examples include marl, mottle and melange yarns, yarns with different levels of hairiness, and yarns produced by various spinning systems. The model can be used for 3D simulation of yarns in computer-aided design systems for textile design and for the prediction of mechanical properties of yarns.