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

Abstract

For a cyclic covering map $(\Sigma ,K)\to ({\Sigma }^{\prime },K^{\prime })$ between two pairs of a 3-manifold and a knot each, we describe the fundamental group ${\pi }_1(\Sigma \setminus K)$ in terms of ${\pi }_1({\Sigma }^{\prime }\setminus K^{\prime })$. 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.

11 pages, no figure

References in corpus (1)

• Noncommutative knot theory

Cited by in corpus (1)

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