Tied links in the solid torus

TL;DR

This paper introduces tied links in the solid torus, generalizes previous concepts, and defines an invariant using skein relations and algebraic methods, expanding the understanding of link invariants in 3-manifolds.

Contribution

It extends the concept of tied links to the solid torus and constructs a new invariant via skein relations and algebraic techniques.

Findings

Defined a new invariant for tied links in the solid torus

Connected the invariant to Jones' method and bt-algebra of type B

Provided a framework for analyzing tied links in 3-manifolds

Abstract

We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and subsequently we recover this invariant by using Jones' method over the bt-algebra of type $\mathtt{B}$ and the Markov trace defined on this.