Algebraic Markov equivalence for links in three-manifolds

Abstract

Let B-n denote the classical braid group on n strands and let the mixed braid group B-m,B-n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let B-m,B-infinity = boolean OR(n) B-m,B-n. We describe explicit algebraic moves on B-m,B-infinity such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-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 a(i). This means premultiplication by a(i)(-1) and postmultiplication by a 'combed' version of a(i). For closed three-manifolds there is an additional set of 'combed' band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using L-moves (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 three-manifolds.