Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. Moreover, the core curve of the filling solid torus in these fillings is a geodesic in the resulting hyperbolic manifold.


Typically the study of exceptional Dehn surgery on hyperbolic 3-manifolds is concerned with the production of non-hyperbolic 3-manifolds. However, it is possible for a hyperbolic manifold to result without being a deformation of the original. For example, a "random" knot in a hyperbolic 3-manifold won't be isotopic to a geodesic though is likely to have hyperbolic complement. The meridional filling of this knot exterior would then be an exceptional hyperbolic filling.


What's the maximum number of possible exceptional hyperbolic fillings? What manifolds realize this maximum?



Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds
Sixty is an upper bound.


Hodgson and Kerckhoff's Universal Hyperbolic Dehn Filling theorem ("Universal bounds for hyperbolic Dehn surgery." Annals of Mathematics. 162(1), 367-421) says that, in a one-cusped manifold, you can push the cone angle up from zero all the way up to $2\pi$ as long as the normalized length of the filling slope is at least $7.515$. This implies that the Dehn filling is a hyperbolic Dehn filling (meaning the core is geodesic) when the slope has normalized length that big. This excludes at most 60 slopes.


