Knot Theory in Doubly Periodic Tangles and Applications

Doubly periodic entangled structures offer an interesting framework for modeling and investigating diverse materials and physical phenomena, from micro to large scales. Specifically, a doubly periodic tangle (DP tangle) is characterized as an embedding of an infinite number of curves in the thickened plane, derived as the lift of a link in the thickened torus to the universal cover. DP tangles play a crucial role in scientific research, particularly in fields such as materials science, molecular chemistry, and biology. Despite their widespread applications, a universally accepted mathematical description of DP tangles is currently lacking. One of the key challenges arises from the infinite possibilities in choosing a periodic cell (referred to as a motif) for a DP tangle, taking into account various periodic boundary conditions. In this presentation, we conduct a comprehensive examination of the concept of topological equivalence of DP tangles, offering insights into potential classifications and applications in the process.