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 $Sigma_g imes [0,1]$, 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 $S^2 imes S^1$.

A Markov theorem for generalized plat decomposition

Cattabriga, Alessia
;
2020

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 $Sigma_g imes [0,1]$, 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 $S^2 imes S^1$.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/779921
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