  In this paper we first give a one-move version of Markov's braid theorem for
knot isotopy in $S^3$ that sharpens the classical theorem. Then a relative
version of Markov's theorem concerning a fixed braided portion in the knot. We
also prove an analogue of Markov's 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.
