Arc diagrams on 3-manifold spines

TL;DR

This paper introduces a combinatorial framework for link projections on 3-manifold spines, proving a Reidemeister-type theorem and connecting it to Turaev's shadow theory, advancing the understanding of link isotopies in 3-manifolds.

Contribution

It develops a new theory of link projections on 3-manifold spines, proves a Reidemeister theorem for these projections, and relates crossingless diagrams to shadow equivalence in 4-manifolds.

Findings

Reidemeister moves for link projections on spines established

Any link admits a crossingless projection on any special spine

Connection made between link projections and Turaev's shadow theory

Abstract

We develop a theory of link projections to trivalent spines of 3-manifolds. We prove a Reidemeister Theorem providing a set of combinatorial moves sufficient to relate the projections of isotopic links. We also show that any link admits a crossingless projection to any special spine and we refine our theorem to provide a set of combinatorial moves sufficient to relate crossingless diagrams. Finally, we discuss the connection to Turaev's shadow world, interpreting our result as a statement about shadow equivalence of a class of 4-manifolds.