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

Yuta Nozaki

doi:10.1142/s0218216518500451

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arXiv · 1602.05884

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

Yuta Nozaki

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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.

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Yuta Nozaki. 2016-02-18. An explicit relation between knot groups in lens spaces and those in $S^3$. https://doi.org/10.1142/s0218216518500451

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