Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in R3 that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus T3=S1×S1×S1 in its universal cover R3, where two non-isotopic links in T3 may possess the same TP tangle preimage. We consider the equivalence of TP tangles in R3 through the use of diagrams representing links in T3. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in T3. The final definition of a tridiagram of a link in T3 allows us to then consider additional notions of equivalence relating non-isotopic links in T3 that possess the same TP tangle preimage.
