  A periodic weave is the lift of a particular link embedded in a thickened
surface to the universal cover. Its components are infinite unknotted simple
open curves that can be partitioned in at least two distinct sets of threads.
The classification of periodic weaves can be reduced to the one of their
generating cells, namely their weaving motifs. However, this classification
cannot be achieved through the classical theory of links in thickened surfaces
since periodicity in the universal cover is not encoded. In this paper, we
first introduce the notion of hyperbolic periodic weaves, which generalizes our
doubly periodic weaves embedded in the Euclidean thickened plane. Then, Tait
First and Second Conjectures are extended to minimal reduced alternating
weaving motifs and proved using a generalized Kauffman bracket polynomial
defined for periodic weaving diagrams of the Euclidean plane and generalized to
the hyperbolic plane. The first conjecture states that any minimal alternating
reduced weaving motif has the minimum possible number of crossings, while the
second one formulates that two such oriented weaving motifs have the same
writhe.
