  A weave is the lift to the Euclidean thickened plane of a set of infinitely
many planar crossed geodesics, that can be characterized by a number of sets of
threads describing the organization of the non-intersecting curves, together
with a set of crossing sequences representing the entanglements. In this paper,
the classification of a specific class of doubly periodic weaves, called
untwisted (p,q)-weaves, is done by their crossing number, which is the minimum
number of crossings that can possibly be found in a unit cell of its infinite
weaving diagrams. Such a diagram can be considered as a particular type of
quadrivalent periodic planar graph with an over or under information at each
vertex, whose unit cell corresponds to a link diagram in a thickened torus.
Moreover, considering that a weave is not uniquely defined by its sets of
threads and its crossing sequences, we also specify the notion of equivalence
classes by introducing a new parameter, called crossing matrix.
