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 intersecting 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) that preserve the decomposition. We compute the Goeritz groups of all (1, 1)-decompositions.