A Markov theorem for plat closure of surface braids in Dunwoody and periodic Takahashi manifolds

In this article we deal with the problem of finding equivalence moves for links in Dunwoody and periodic Takahashi manifolds. We represent these manifolds using Heegaard splitting and we represent the embedded links as plat closure of elements in the braid group of the corresponding Heegaard surfaces. More precisely, starting from an open Heegaard diagram for such manifolds, we determine the plat slide equivalence moves algorithmically and compute them explicitly in some cases.