Cyclic branched cover of a knot

Assume the Axiom of Choice. Let $K$ be a smooth oriented knot in the standard oriented $3$-sphere, choose its compact exterior $E_K$ with boundary torus $T_K$, and let $\mu$ and $\lambda$ be its positive meridian and preferred longitude. The exterior homology theorem gives $H_1(E_K;\mathbb{Z})=\mathbb{Z}[\mu ]$. Let $\varphi :{\pi }_1(E_K)\to \mathbb{Z}$ be the composite of abelianization with the homomorphism $[\mu ]\mapsto 1$. For an integer $q\ge 2$, let ${\varphi }_q$ be the composite ${\pi }_1(E_K)\stackrel{\varphi }{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}q\mathbb{Z}$, and let $p_q:E_q\to E_K$ be the connected cover classified by $\ker {\varphi }_q$. On $T_K$, the homomorphism sends $[\mu ]\mapsto 1$ and $[\lambda ]\mapsto 0$, so the preimage of $T_K$ is a connected torus cover whose fundamental group corresponds to $⟨{\mu }^q,\lambda ⟩$. Let $\tilde{\mu }$ be the closed lift of ${\mu }^q$. Attach a solid torus to $E_q$ with its meridian identified with $\tilde{\mu }$; the resulting oriented closed $3$-manifold is the $q$-fold cyclic branched cover ${\mathrm{\Sigma }}_q(K)$. In tubular coordinates near $K$, the extended map to $S^3$ has local form $(s,w)\mapsto (s,w^q)$, and the deck group $\mathbb{Z}\mathrm{/}q\mathbb{Z}$ acts by $w\mapsto e^{2\pi i\mathrm{/}q}w$.

Smooth oriented knots in the standard oriented three-sphere

The chosen compact exterior and boundary torus

The exterior is a smooth manifold with torus boundary and retracts from the open complement

Connectedness and local covering hypotheses for the knot complement

Positive meridian and preferred longitude conventions

The positive meridian generates exterior homology and the longitude is null-homologous

The first homology of the exterior is the abelianization of its fundamental group

Covering maps and their sheets

Connected covers classified by fundamental-group subgroups

Definition of a regular covering

A normal subgroup gives a regular covering

The deck group is the quotient by the covering subgroup

Deck transformations and deck group

The Axiom of Choice used by the exterior-homology theorem