We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Delm fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: for every integer n, we can prove that there are infinitely many hyperbolic knots in S-3 having exceptional surgeries {n, n + 1, n + 2, n + 3}, with n + 1, n + 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds. we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements. we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements. we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling.
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