Markovʹs theorem in 3-manifolds

In this paper we first give a one-move version of Markovʹs braid theorem for knot isotopy in S3 that sharpens the classical theorem. Then we give a relative version of Markovʹs theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markovʹs theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.