On 3-periodic tangled networks and their untangling number

Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. A single or multiple networks can have different configurations in space, where entanglement may arise due to the way the edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we describe a diagrammatic representation of 3-periodic tangled networks that allows the definition of the untangling number, which can be used as a measure of entanglement complexity. The untangling number, first defined elsewhere for 3-periodic entanglements of filaments and reminiscent of the unknotting number in knot theory, quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state. This distance is measured through a sequence of operations called untangling operations performed on a diagram of a given structure.