A Markov theorem for generalized plat decomposition

TL;DR

This paper establishes a Markov theorem for tame links in 3-manifolds using plat-like representations and surface braid groups, providing algebraic criteria for link equivalence in such manifolds.

Contribution

It introduces a Markov theorem for links in 3-manifolds via surface braid groups and analyzes isotopy equivalences through algebraic and geometric methods.

Findings

Provides explicit constructions for genus 1 manifolds like lens spaces and S^2×S^1.

Translates link isotopy in 3-manifolds into algebraic equivalence in surface braid groups.

Analyzes the effect of sliding isotopies on braid representatives.

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 $S^2\times S^1$.