TITLE: What constant ensures hyperbolicity of Dehn surgery?
QUESTION [9 upvotes]: I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from reading some papers, it seems that the following is correct (so the first question - is it correct?). Given a hyperbolic link, there is a constant, such that filling a subset of components with slopes with denominators bigger than this given constant, one always gets a hyperbolic manifold (complete, finite volume?).
So my real question is: can we explicitly find this constant? We may or may not be given the volume of the initial link.

REPLY [13 votes]: Just a remark: there is no universal such constant, even for links with bounded volume complement. 
For example, there are infinitely many links with the same complement as the Whitehead link, e.g. 

Then $\frac14$ Dehn filling on the blue component gives the trivial knot,
which is not hyperbolic (at least not with finite volume). Adding twists
to the red component, we may obtain examples with arbitrarily large 
denominator non-hyperbolic Dehn fillings.