An explicit relation between knot groups in lens spaces and those in $S^3$

TL;DR

This paper establishes a direct relationship between the fundamental groups of knots in lens spaces and those in the 3-sphere, providing new insights into knot representations and free periods.

Contribution

It offers an explicit description of knot groups under cyclic covers and presents an alternative proof regarding the non-representability of certain knots in lens spaces as preimages.

Findings

Derived a formula relating knot groups in lens spaces to those in $S^3$

Proved that some knots in $S^3$ cannot be preimages of knots in lens spaces

Highlighted the role of subgroup generated by commutators and pth powers in proofs

Abstract

For a cyclic covering map $(\Sigma,K) \to (\Sigma',K')$ between two pairs of a 3-manifold and a knot each, we describe the fundamental group $\pi_1(\Sigma \setminus K)$ in terms of $\pi_1(\Sigma' \setminus K')$. As a consequence, we give an alternative proof for the fact that certain knots in $S^3$ cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group $G$ generated by the commutators and the $p$th power of each element of $G$ plays a key role.