Communications in Analysis and Geometry

Volume 14 (2006)

Number 5

Dehn Filling of the "Magic" 3-manifold

Pages: 969 – 1026



Bruno Martelli

Carlo Petronio


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 Dehn 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 S3 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.

Published 1 January 2006