What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm particularly interested in the case where the conical angle $\alpha$ around $K$ is less than $2\pi$.

The classic example of this is the figure-8 knot. If you quotient $\mathbb E^3$ by the crystallographic group $\mathrm P2_13$, you get a Euclidean *orbifold* structure on $S^3$ whose singular locus is the figure-8 knot with angle $2\pi/3$. Dunbar (pp.82–86) gives a classification of all Euclidean orbifold structures on $S^3$, and this is the only one whose singular locus is a knot. Therefore, we know that if $K$ is not the figure-8 knot, then $\alpha$ cannot equal $2\pi/n$. That's a starting point, at least!

*Dunbar, William D.*, **Geometric orbifolds**, Rev. Mat. Univ. Complutense Madr. 1, No. 1-3, 67-99 (1988). ZBL0655.57008.

**Edit.** Mednykh and Rasskazov construct a very simple family of fundamental polyhedra $\mathcal P(\theta)$ for the polyhedral manifolds $\mathcal C(\theta)$ ($S^3$ with conical angle $\theta$ along the figure-8 knot), for $\theta\in[0,4\pi/3)$. They show that, as $\theta$ increases, $\mathcal P(\theta)$ goes from hyperbolic to spherical, passing through Euclidean at $\theta=2\pi/3$. The reason I mention this is that they note (p.446):

...that [their] approach is general and can be applied to other two-bridge links and knots.

And again, they (p.448):

...describe an algorithm for the construction of the fundamental set $\mathcal P$ for the figure-eight orbifold which effectively works in every space of constant sectional curvature. With slight modifications this algorithm can also be used to construct the fundamental set of any 2-bridge link orbifold.

*Mednykh, Alexander; Rasskazov, Alexey*, **Volumes and degeneration of cone-structures on the figure-eight knot**, Tokyo J. Math. 29, No. 2, 445-464 (2006). ZBL1124.57008.

**Final Edit.** According to a recent paper of Mednykh, the two-bridge knots $4_1,5_2,6_1,6_2,6_3,7_2,7_3,7_4,7_5,7_6$ and $7_7$ all admit polyhedral metrics with a given conical angle $\alpha_0<\pi$. With the exception of $7_5,7_6$ and $7_7$, the value of $\alpha_0$ is given implicitly, but exactly.

Do all two-bridge knots have this property?