A Markov theorem for generalized plat decomposition

Abstract

We prove a Markov theorem for tame links in a connected closed orientable 3-manifold $M$ with respect to a plat-like representation. More precisely, given a genus $g$ Heegaard surface ${\Sigma }_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}={\pi }_1(C_{2n}({\Sigma }_g))$ and analyze how to translate the equivalence of links in $M$ under ambient isotopy into an algebraic equivalence in $B_{g,2n}$. First, we study the equivalence problem in ${\Sigma }_g\times [0,1]$, and then, to obtain the equivalence in $M$, we investigate how isotopies corresponding to "sliding" along meridian discs change the braid representative. At the end we provide explicit constructions for Heegaard genus 1 manifolds, i.e. lens spaces and $S^2\times S^1$.

Acknowledgements added. Accepted for publication on Ann. Sc. Norm. Super. Pisa Cl. Sci