Exceptional Slopes on Manifolds of Small Complexity

TL;DR

This paper classifies all exceptional slopes and fillings for hyperbolic 3-manifolds obtained from the minimally twisted 5-chain link, advancing understanding of small complexity manifolds and supporting conjectures.

Contribution

It provides a complete classification of exceptional slopes and fillings for surgeries on the 5-chain link, covering many small complexity hyperbolic manifolds.

Findings

Classified exceptional slopes for all surgeries on the 5-chain link

Described exceptional fillings for various small complexity manifolds

Identified an infinite family with specific Seifert and toroidal fillings

Abstract

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.