The untangling number of 3-periodic tangles
Abstract
The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry.
Keywords
Cite
@article{arxiv.2504.01747,
title = {The untangling number of 3-periodic tangles},
author = {Toky Andriamanalina and Sonia Mahmoudi and Myfanwy E. Evans},
journal= {arXiv preprint arXiv:2504.01747},
year = {2026}
}