For the purposes of this
paper, Dehn surgery along a curve K in a 3-manifold M with slope σ is ‘exceptional’
if the resulting 3-manifold MK(σ) is reducible or a solid torus, or the core of the
surgery solid torus has finite order in π1(MK(σ)). We show that, providing
the exterior of K is irreducible and atoroidal, and the distance between
σ 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
∂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.
