On the Kauffman bracket skein module of surgery on a trefoil

We estimate the rank of the Kauffman bracket skein module of each manifold obtained from integral surgery on a trefoil knot. It is well known that all but two of these manifolds contain no incompressible surfaces. We find that the two exceptions are exactly those whose skein module is not finitely generated, thereby extending a pattern that holds for all known compact orientable examples.