ar X iv : m at h / 04 05 49 3 v 3 [ m at h . G T ] 1 6 Fe b 20 06 ALGEBRAIC MARKOV EQUIVALENCE FOR LINKS IN 3 – MANIFOLDS

Let Bn denote the classical braid group on n strands and let the mixed braid group Bm,n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let Bm,∞ = ∪nBm,n. We will describe explicit algebraic moves on Bm,∞ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented 3–manifold. The moves depend on a fixed link representing the manifold in S 3. More precisely, for link complements the moves are: the two familiar moves of the classical Markov equivalence together with 'twisted' conjugation by certain loops ai. This means premultiplication by ai −1 and post-multiplication by a 'combed' version of ai. For closed 3–manifolds there is an additional set of 'combed' band moves which correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov Theorem using L–moves [12] (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov Theorem that classifies links in S 3 up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of 3–manifolds.