The untangling number of 3-periodic tangles

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.