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.