Double branched cover of an oriented link

Assume the Axiom of Choice. Let $L=K_1\cup \cdots \cup K_r$ be a nonempty finite oriented smooth link in the standard oriented three-sphere, and put $X_L=S^3\setminus L$. A small meridian about $K_i$ is the boundary of a normal disk fiber; its orientation is immaterial modulo two. The mod-two meridian character is the homomorphism ${ϵ}_L:{\pi }_1(X_L)\to \mathbb{Z}\mathrm{/}2$ that sends every component meridian to $1$. For each $K_i$, choose an oriented tubular chart, meaning a diffeomorphism from a normal disk bundle onto a neighborhood, written ${\phi }_i:(\mathbb{R}\mathrm{/}2\pi \mathbb{Z})\times D^2\to N_i$ and respecting the core orientation. Let ${\lambda }_i$ be the parallel longitude given by ${\phi }_i((\mathbb{R}\mathrm{/}2\pi \mathbb{Z})\times \{z_0\})$ for a fixed $z_0\in \mathrm{\partial }{D}^2$, and set ${\eta }_i={ϵ}_L([{\lambda }_i])\in \{0,1\}$. The double branched cover $\mathrm{\Sigma }(L)$ is the closed three-manifold obtained by taking the connected two-sheeted cover of $X_L$ classified by $\ker {ϵ}_L$ and extending it over each $K_i$ with local normal-disk model $(s,z)\mapsto (s,e^{\mathrm{i}{\eta }_{i}s}{z}^2)$ in these coordinates. Give its total space the orientation for which the covering map is orientation-preserving away from the branch locus.

Finite oriented smooth links in the standard three-sphere

The complement is an eligible connected manifold

Connected covers and fundamental-group subgroups

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, §§1–2

Hatcher, Algebraic Topology, Theorem 1.38 and Proposition 1.39