Braid equivalence in 3-manifolds with rational surgery description

TL;DR

This paper develops a comprehensive framework for understanding braid equivalence in 3-manifolds obtained via rational surgery, providing geometric and algebraic tools to classify knots and links in these spaces.

Contribution

It introduces a sharpened Reidemeister theorem and algebraic formulations for braid equivalence in 3-manifolds, extending the mixed braid group approach.

Findings

Provides geometric formulations of braid equivalence using mixed braids and L-moves.

Offers algebraic descriptions of braid equivalence in mixed braid groups.

Applies the framework to specific 3-manifolds like lens spaces and Seifert manifolds.

Abstract

In this paper we describe braid equivalence for knots and links in a 3-manifold $M$ obtained by rational surgery along a framed link in $S^3$. We first prove a sharpened version of the Reidemeister theorem for links in $M$. We then give geometric formulations of the braid equivalence via mixed braids in $S^3$ using the $L$-moves and the braid band moves. We finally give algebraic formulations in terms of the mixed braid groups $B_{m,n}$ using cabling and the techniques of parting and combing for mixed braids. We also provide concrete formuli of the braid equivalence in the case where $M$ is a lens space, a Seifert manifold or a homology sphere obtained from the trefoil. The algebraic classification of knots and links in a $3$-manifold via mixed braids is a useful tool for studying skein modules of $3$-manifolds.