Links in the spherical 3-manifold obtained from the quaternion group and their lifts

TL;DR

This paper constructs infinite families of hyperbolic links in lens spaces with non-isotopic links whose lifts to the 3-sphere are isotopic, using quaternion group symmetries and diagrammatic methods.

Contribution

It introduces a method to generate hyperbolic links in lens spaces with isotopic lifts, expanding understanding of link behavior under covering spaces.

Findings

Existence of infinitely many non-isotopic hyperbolic links with isotopic lifts

Development of a diagrammatic approach using square projections

Identification of Reidemeister-type moves connecting isotopic diagrams

Abstract

We show that there are infinitely many triples of non-isotopic hyperbolic links in the lens space $L(4,1)$ such that the three lifts of each triple in $S^{3}$ are isotopic. They are obtained as the lifts of links in $S^{3} / Q_{8}$ by double covers, where $Q_{8}$ is the quaternion group. To construct specific examples, we introduce a diagram of a link in $S^{3} / Q_{8}$ obtained by projecting to a square. The diagrams of isotopic links are connected by Reidemeister-type moves.