14
$\begingroup$
  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?

$\endgroup$
14
$\begingroup$

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 $K$.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ – Giulio Belletti Oct 10 '18 at 13:33
  • $\begingroup$ Yes, definitely. I am editing my answer accordingly. $\endgroup$ – Roberto Frigerio Oct 10 '18 at 14:01
6
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.