Communications in Analysis and Geometry
Volume 22 (2014)
Exceptional Dehn surgery on the minimally twisted five-chain link
Pages: 689 – 735
We consider in this paper the minimally twisted chain link with five components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior $M_5$. The 3-manifold $M_5$ is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (including 0) cusps. In view of Thurston’s hyperbolic Dehn filling theorem it is then natural to face the problem of classifying all the exceptional fillings on $M_5$, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the symmetries of $M_5$ and of some hyperbolic manifolds resulting from fillings of $M_5$, the set of exceptional fillings on $M_5$ is described by a very small amount of information.