The Goeritz groups of (1, 1) -decompositions

A (g, n) -decomposition of a link L in a closed orientable 3-manifold M is a decomposition of M by a closed orientable surface of genus g into two handebodies each of which intersects the link L in n trivial arcs. The Goeritz group of that decomposition is then defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the pair (M, L) preserving the decomposition. We compute the Goeritz groups of all (1, 1) -decompositions.