A calculus for ideal triangulations of three-manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three-manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,alpha), where M is a three-manifold and alpha is a collection of properly embedded arcs. We also show that certain well-understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,alpha). Our proof does not assume the Matveev-Piergallini calculus for ideal triangulations, and actually easily implies this calculus.