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.

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.

Typical examples of textile structures are separated by finite type invariants of knots in non-trivial (in particular, non-orientable) manifolds. A new series of such invariants is described.

The paper addresses the enumeration problem for k-tangles. We introduce the notion of a cascade diagram of a k-tangle projection and suggest an effective enumeration algorithm for projections based on the cascade representation. Tangle projections and alternating tangles with up to 12 crossings are tabulated. We also provide pictures of alternating k-tangles with at most five crossings.