Communications in Analysis and Geometry

Volume 22 (2014)

Number 4

Exceptional Dehn surgery on the minimally twisted five-chain link

Pages: 689 – 735



Bruno Martelli (Dipartimento di Matematica, Università di Pisa, Italy)

Carlo Petronio (Dipartimento di Matematica, Università di Pisa, Italy)

Fionntan Roukema (School of Mathematics and Statistics, University of Sheffield, United Kingdom)


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.

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