Diagrammatic representations of 3-periodic entanglements

Diagrams enable the use of various algebraic and geometric tools in analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures, which are embeddings of simple curves in R³ that are invariant under translations along three non-coplanar axes. These diagrams require an extended set of new moves in addition to the Reidemeister moves, which we show to preserve ambient isotopies of triply periodic entangled structures. We use the diagrams to define the crossing number and the unknotting number of the triply periodic entanglements, demonstrating the practicality of the diagrammatic representation.