  Doubly periodic weaves and polycatenanes embedded in the thickened Euclidean
plane are three-dimensional complex entangled structures whose topological
properties can be encoded in any generating cell of its infinite planar
representation. Such a periodic cell, called motif, is a specific type of link
diagram embedded on a torus consisting of essential simple closed curves for
weaves, or null-homotopic for polycatenanes. In this paper, we introduce a
methodology to construct such motifs using the concept of polygonal link
transformations. This approach generalizes to the Euclidean plane existing
methods to construct polyhedral links in the three-dimensional space. Then, we
will state our main result which allows one to predict the type of motif that
can be built from a given planar periodic tiling and a chosen polygonal link
method.

Geometric Topology