# Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?

1. 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?
2. What if in Question 1 we restrict the cone angles to be $$\leq \pi$$?
3. 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?

The results proved in

S. Kojima, "Deformations of hyperbolic 3-cone-manifolds", J. Differential Geom. 49 (1998), no. 3, 469-516

provide complete answers to questions 1 and 3.

The main theorem of the cited paper ensures that any compact hyperbolic cone metric with cone angles at most π can be continuously deformed to a complete hyperbolic metric on the complement of the singularity. Furthermore this deformation can be done by an angle decreasing deformation.

This provides a partial answer to your question (2) and a complete answer to (3): If $$K$$ is the singular locus of a cone structure with angles $$\leq \pi$$, then $$K$$ is necessarily hyperbolic. Moreover, thanks e.g. to a suitable version of the Schlafli formula, the hyperbolic volume of the cone structure is stricly smaller than the hyperbolic volume of the cusped manifold $$S^3\setminus K$$, which in turn is equal to the simplicial volume of $$S^3\setminus K$$ (up to the constant $$v_3$$, which I guess you forgot to mention when asking question 3).

Moreover, Theorem 1.2.1 of Kojima's paper implies that, if $$S^3$$ supports a cone structure with singularity $$K$$, then $$K$$ must be hyperbolic, regardless of the cone angle of the original cone structure along $$K$$.

• Thanks for your answer! By glancing through the paper you mentioned, it seems to me that Theorem 1.2.1 (page 474) would also give a complete answer to question 1: K must be hyperbolic regardless of any angle assumption. Oct 10, 2018 at 13:33
• Yes, definitely. I am editing my answer accordingly. Oct 10, 2018 at 14:01

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).