An algorithm for finding minimal volume hyperbolic links and the Dehn parental test

We describe an algorithm that, given a 3-manifold M, outputs a finite set containing all minimal volume k-component hyperbolic link complements in M. A key step, that might be of independent interest, is an algorithm that, given two 3-manifolds N and M, decides whether they are related by Dehn filling. In fact, we show that the set of boundary slopes giving a Dehn filling of N to M is determined by a special class of well-studied quadratic Diophantine (in)equalities, for which solvability is known to be decidable.