Classification of doubly periodic untwisted (p, q) -weaves by their crossing number and matrices

Abstract

A weave is the lift to the thickened Euclidean plane of a particular type of quadrivalent planar connected graph with an over or under crossing information to each vertex and such that the lifted components are non-intersecting simple open curves. In this paper, we introduce a formal topological definition of weaves as three-dimensional entangled structures and characterize the equivalence classes of doubly periodic untwisted (p,q)-weaves by introducing a new invariant, called crossing matrix. Finally, we suggest a combinatorial approach to classify this specific class of weaves by their crossing number.