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A periodic weave is the lift of a particular link embedded in a thickenedsurface to its universal cover. Its components are infinite unknotted simpleopen 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 theirgenerating cells, namely their weaving motifs. However, this cannot be achievedthrough the classical theory of links in a thickened surface since periodicityin the universal cover is not encoded. In this paper, the Tait first and secondconjectures are extended to minimal reduced alternating weaving motifs. Thefirst one states that any minimal alternating reduced weaving motif has theminimum possible number of crossings, while the second one formulates that twosuch oriented weaving motifs have the same writhe.

The Tait First and Second Conjectures for Alternating Periodic Weaves