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

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