Diagrammatic representations of 3-periodic entanglements
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 the 3-dimensional space that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as lifts of links in the 3-torus, where two non-isotopic links in the 3-torus may lift to the same TP tangle. We consider the equivalence of TP tangles in the 3-space through the use of diagrams representing links in the 3-torus. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in the 3-torus. The final definition of a tridiagram of a link in the 3-torus allows us to then consider additional notions of equivalence that encode global isotopies of a TP tangle, such as a change of basis or lattice in the 3-space.
Toky Andriamanalina, Myfanwy E. Evans, and Sonia Mahmoudi
