Braid equivalences in 3-manifolds with rational surgery description

In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M obtained by rational surgery along a framed link in S-3. We do this by representing M by a closed framed braid in 53 and links in M by closed mixed braids in S-3. We first prove an analogue of the Reidemeister theorem for links in M. We then give geometric formulations of the mixed braid equivalence using the L-moves and the braid band moves. Finally we formulate the algebraic braid equivalence in terms of the mixed braid groups B-m,B-n, using cabling and the parting and combing techniques for mixed braids. Our results set a homogeneous algebraic ground for studying links in 3-manifolds and in families of 3-manifolds using computational tools. We provide concrete formuli of the braid equivalences in lens spaces, in Seifert manifolds, in homology spheres obtained from the trefoil and in manifolds obtained from torus knots.Our setting is appropriate for constructing Jones-type invariants for links in families of 3-manifolds via quotient algebras of the mixed braid groups B-m,B-n, as well as for studying skein modules of 3-manifolds, since they provide a controlled algebraic framework and much of the diagrammatic complexity has been absorbed into the proofs. Further, our moves can be used in a braid analogue of Rolfsen's rational calculus and potentially in computing Witten invariants. (C) 2015 Elsevier B.V. All rights reserved.