Equivalence of Doubly Periodic Tangles

TL;DR

This paper develops a comprehensive topological framework for doubly periodic tangles, analyzing their isotopies and equivalences, and extends the theory to various related diagrammatic categories for potential new applications.

Contribution

It provides the first complete mathematical characterization of DP tangle isotopies and introduces generalized categories for broader modeling applications.

Findings

Characterization of DP isotopy as an equivalence relation on motifs

Analysis of local and global isotopies of DP tangles

Extension of the framework to framed, virtual, welded, and other DP tangle categories

Abstract

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.