One natural choice (if not necessarily the best choice) is to pick a small $\epsilon$, take the $\epsilon$-neighbourhood in the three-sphere, and stereographically project that to euclidean space. To actually do this requires a bit of spherical trig.

This is what is done in the paper you cite.

One natural choice (if not necessarily the best choice) is to pick a small $\epsilon$, take the $\epsilon$-neighbourhood in the three-sphere, and stereographically project that to euclidean space. To actually do this requires a bit of spherical trig. Note also that there is no "need" to work in $S^3$. You can instead work with the spherical metric on $\mathbb{R}^3$.