On the isotopies of tangles in periodic 3-manifolds using finite covers

TL;DR

This paper investigates the isotopy classes of periodic tangles in 3-manifolds using finite covers, establishing conditions under which isotopic lifts imply isotopic links.

Contribution

It proves that isotopic lifts in a common finite cover imply the original links are isotopic, advancing understanding of periodic tangles in 3-manifold topology.

Findings

Isotopic lifts in a finite cover imply the original links are isotopic.

Techniques from 3-manifold topology are used to study link complements.

The work relates periodic tangles to their lifts in finite covers.

Abstract

A periodic tangle is a one-dimensional submanifold in $\mathbb{R}^3$ that has translational symmetry in one, two or three transverse directions. A periodic tangle can be seen as the universal cover of a link in the solid torus, the thickened torus, or the three-torus, respectively. Our goal is to study equivalence relations of such periodic tangles. Since all finite covers of a link lift to the same periodic tangle, it is necessary to prove that isotopies between different finite covers are preserved. In this paper, we show that if two links have isotopic lifts in a common finite cover, then they are isotopic. To do so, we employ techniques from 3-manifold topology to study the complements of such links.