Diagrams of links and bands on 3-manifold spines and flow-spines

Abstract

The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In this note we replace $3$-space by any 3-manifold $M$ and we extend the Reidemeister theorem (definition of the decoration and description of the combinatorial moves) in four situations, taking diagrams either of links or of bands (collections of cylinders and M\"obius strips), either on an almost special spine or on a flow-spine of $M$. This partially reproves and extends a result of Brand, Burton, Dancso, He, Jackson and Licata.