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.