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

TL;DR

The paper investigates links in the quotient $X=S^{3}/Q_{8}$ and their lifts to the lens space $L(4,1)$ via the double cover structure. It develops a cube-to-square diagrammatic framework for $S^{3}/Q_{8}$ and proves that isotopy classes are connected by generalized Reidemeister moves $R_{1}-R_{8}$, enabling controlled construction of lifted links. The authors prove the existence of infinitely many triples of non-isotopic hyperbolic links in $L(4,1)$ whose three lifts to $S^{3}$ are isotopic, providing explicit Seifert-fibered and hyperbolic examples; they further analyze the symmetry actions and arithmetic properties of the resulting manifolds. This work advances understanding of link lifts across nontrivial 3-manifold coverings and offers concrete diagrammatic tools to engineer high-volume hyperbolic links via covers, with implications for the study of freely periodic symmetries and Seifert fibrations in lens spaces.

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.

Figures (19)

• Figure 1: The fundamental domain $C$
• Figure 2: A diagram of a link in $S^{3} / Q_{8}$
• Figure 3: Generalized Reidemeister moves
• Figure 4: An $R_{8}$ move as a composition of $R_{8}$ and $R_{6}$ moves
• Figure 5: The tangle $T_{n}$ and the manifold $N_{n}$

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Remark 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof