Diagrammatic representations of 3-periodic entanglements

Abstract: 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 $\mathbb{R}<sup>3$ 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 $$\mathbb{T}³ = \mathbb{S}¹ \times \mathbb{S}¹ \times \mathbb{S}<sup>1$$ in its universal cover $\mathbb{R}<sup>3$, where two non-isotopic links in $\mathbb{T}<sup>3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}<sup>3$ through the use of diagrams representing links in $\mathbb{T}<sup>3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}<sup>3$. The final definition of a tridiagram of a link in $\mathbb{T}<sup>3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}<sup>3$ that possess the same TP tangle preimage.