2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71330In 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.29 pages, 41 figures, LaTex documentGeometric Topology57M25text