Tied Links in various topological settings

TL;DR

This paper extends the concept of tied links to various 3-manifolds, introduces algebraic structures for their study, and establishes foundational theorems like Alexander's and Markov's for these settings.

Contribution

It generalizes tied links to new topological spaces and develops the algebraic framework and key theorems for their analysis.

Findings

Defined tied braid groups in new topological contexts

Proved Alexander's theorem for tied links in these manifolds

Proved Markov's theorem for tied links in these settings

Abstract

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called {\it ties}, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid groups $TB_{g, n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we state and prove Alexander's and Markov's theorems for tied links in the 3-manifolds mentioned above. Finally, we emphasize on further steps needed in order to study tied links in knot complements and c.c.o. 3-manifolds, which is the subject of a sequel paper.