Exceptional surgery curves in triangulated 3-manifolds

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundary of M a (possibly empty) union of tori. Moreover, there is a simple algorithm to find all these surgery curves, which involves inserting tangles into the 3-simplices of any given triangulation of M. As a consequence, we deduce some results about the finiteness of certain unknotting operations on knots in the 3-sphere.