Convex hulls and isometries of cusped hyperbolic 3-manifolds

Abstract

An algorithm for computing canonical triangulations of cusped hyperbolic 3-manifolds provides an efficient way to determine whether two such manifolds are isometric. The canonical triangulation is defined via a convex hull construction in Minkowski space. The algorithm accepts as input an arbitrary triangulation (which typically corresponds to a nonconvex solid in Minkowski space) and locally modifies it until it arrives at the canonical triangulation (which corresponds to the convex hull). The practicality of the algorithm rests on a surprisingly simple theorem which detects where the local modifications must be made. The algorithm has found many applications; for example, it quickly determines whether two hyperbolic knots are equivalent.