TITLE: Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
QUESTION [14 upvotes]: Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $\leq \pi$?
Is it true that, if $M$ is a cone-manifold as in question 2, its volume is at most the simplicial volume of $S^3-K$ times $v_3$?

I would also appreciate partial answers: it is known of some knots that aren't singularities of hyperbolic cone-manifold structures? Is it known of some non-hyperbolic knots that are? What is known about their volume?

REPLY [6 votes]: Let me add some remarks in addition to Roberto's answer. 
Question 2. has the same answer as Question 1: a knot is the singular locus of a hyperbolic cone metric iff the knot has hyperbolic complement (complete finite-volume hyperbolic metric) iff there is a cone metric with angle $\leq \pi$. This follows from Thurston's hyperbolic Dehn filling theorem: the complete metric may be thought of as a metric with cone angle zero, and the angle may be perturbed to be $\epsilon < \pi$. 
For 3., the volume relation holds whether or not the cone angles are $\leq \pi$. When the angles are $\leq \pi$, Kojima shows that the cone angle may be continuously deformed to $0$, and that the volume increases during this deformation by Schlafli's formula. As far as I know, an analogous deformation result is still unknown for cone angles $\geq \pi$. However, there is a global comparison result which implies that the volume is at most the simplicial volume of the knot complement. This follows from the "natural map" technique of Besson-Courtois-Gallot. One may also prove this using Gromov's approach of measurable cycles and simplicial volume (Ben Klaff did this in his thesis).