Kauffman-Type Polynomial Invariants for Doubly Periodic Structures

Abstract

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.

Classification of Knots of Small Complexity in Thickened Tori

“…We show that the obtained knots are different with the help of knot invariants that are counterparts of the Kauffman polynomial invariants [9] (cf. also [8]). …”

“…A nontrivial loop can be added (in a unique way) to the projections 3 4 and 3 5 . This yields the projections 4 7 and 4 8 . In addition, the projections 4 9 and 4 10 of type l, which can serve as the projections of prime knots, are obtained by adding nontrivial loops to the unique projection of type g which is the projection of a composite knot (a nontrivial circle with a local trefoil).…”

“…At present, all the knots in S 3 such that their whose minimal diagrams have at most 16 crossing points are tabulated [4]. We also mention the works [5,6] devoted to knots in a projective space and [7,8] devoted to classifications of tangles.…”

Invariants of Links in 3-Manifolds and Splitting Problem of Textile Structures

“…For some different constructions of link invariants in manifolds M 2 × R 1 and closely related virtual link invariants, mainly developing the Kauffman and Khovanov constructions, see [18], [19] and references therein, and also [10]. + and s ′ − the results of our isotopy J applied to s + and s − respectively, in particular s ′…”

Doubly Periodic Textile Structures

“…The Alexander data for Leno weave is presented in Figure 13, where Figure 14. It is based on the choice of generators in Figure 8, and the corresponding Alexander polynomial in equation (2). In this case the data form is given simply by U = y, V = x.…”