2 6 M ay 2 00 4 Markov ’ s theorem in 3 – manifolds

In this paper we first give a one-move version of Markov’s braid theorem for knot isotopy in S 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. 1 Overview According to Birman [3], Markov [13] originally stated his braid equivalence theorem using three braid moves; later there was another brief announcement of an improved version of Markov’s theorem by Weinberg [23] consisting of the two well-known braid-equivalence moves: conjugation in the braid groups and the ‘stabilizing’ or ‘Markov’ moves (M–moves). Our first main result is a one-move Markov theorem which we now state. An L–move on a braid consists of cutting one arc of the braid open and splicing into the broken strand new strands to the top and bottom, both either under or over the rest of the braid: figure 1 breakpoint L-under L-over new strands new strands L–moves and isotopy generate an equivalence relation on braids called L–equivalence. In §4 we prove that L–equivalence classes of braids are in bijective correspondence with isotopy classes of oriented links in S, where the bijection is induced by ‘closing’ the braid to form a link. As a consequence, L–equivalence is the same as the usual Markov equivalence and thus the classical Markov theorem is sharpened. The proofs are based on a canonical