Communications in Analysis and Geometry

Volume 27 (2019)

Number 5

Exceptional slopes on manifolds of small complexity

Pages: 1105 – 1161



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


It has been observed that most manifolds in the Callahan–Hildebrand–Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted $5$-chain link. A full classification of the exceptional surgeries on the $5$-chain link has recently been completed. In this article, we provide a complete classification of the sets of exceptional slopes and fillings for all cusped hyperbolic surgeries on the minimally twisted $5$-chain link, thereby describing the sets of exceptional slopes and fillings for most hyperbolic manifolds of small complexity. The classification produces the description of exceptional fillings for many families of one and two cusped manifolds, and provides supporting evidence for some well-known conjectures. One such family that appears in the classification is an infinite family of $1$-cusped hyperbolic manifolds with four Seifert manifold fillings and a toroidal filling.

Received 30 October 2015

Accepted 20 April 2017

Published 12 November 2019