  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.
