Exceptional Dehn surgery on the minimally twisted five-chain link

TL;DR

This paper classifies all exceptional Dehn fillings on the minimally twisted five-chain link exterior, revealing a small set of non-hyperbolic manifolds due to symmetries and hyperbolic geometry.

Contribution

It provides a complete classification of exceptional Dehn surgeries on the minimally twisted five-chain link, leveraging symmetries and hyperbolic geometry to simplify the analysis.

Findings

Complete list of exceptional fillings on M5

Symmetry reduces the complexity of classification

Most fillings yield hyperbolic 3-manifolds

Abstract

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 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 M5, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the symmetries of M5 and of some hyperbolic manifolds resulting from fillings of M5, the set of exceptional fillings on M5 is described by a very small amount of information.