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

TL;DR

This paper extends the Reidemeister theorem to links and bands on various spines of 3-manifolds, providing a combinatorial framework for link diagrams beyond 3-space.

Contribution

It generalizes the Reidemeister theorem to 3-manifolds using diagrams on spines and flow-spines, including links and bands, extending prior results.

Findings

Reidemeister theorem extended to 3-manifolds

Diagrams of links and bands on spines are equivalent under specific moves

Partial reproof and extension of previous results by other researchers

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.