Equivalence of Doubly Periodic Tangles

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots and links in the thickened torus, and which can be chosen in infinitely many ways. DP tangles are used in modeling materials and physical systems of entangled filaments. In this paper, we establish the complete mathematical framework of the topological theory of DP tangles. We present an exhaustive analysis of DP tangle isotopies. These are distinguished in local isotopies and global isotopies. Our analysis yields the characterization of DP isotopy as an equivalence relation on the level of their (flat) motifs, called DP tangle equivalence. Along the way, we also discuss motif minimality. We further generalize our results to other diagrammatic categories, namely framed, virtual, welded, singular, pseudo, tied and bonded DP tangles, which could be used in novel applications.