Ribbon graphs in 3-manifolds

Ribbon graphs in a 3-manifold M generalize framed knots and links in M by allowing free ends lying in \(\partial \mathrm{M}\) and rectangular vertices lying in \(\mathrm{Int}(M)\;=\;M\partial M\). We define ribbon graphs in terms of so-called plexuses. Then we explain how to present ribbon graphs by diagrams on skeletons of the manifold and introduce local moves on the diagrams preserving the associated ribbon graphs. The main results of the chapter (Theorems 14.3 and 14.4) claim that any diagrams of isotopic ribbon graphs may be related by our moves.