Principal Congruence Links: Class Number Greater than 1

ABSTRACT

In a previous article, we started an enumeration of the finitely many levels for which a principal congruence manifold can be a link complement in S³. In this article we give a complete enumeration of all the principal congruence link complements in S³, together with their levels in the case when the class number of $\mathbf{Q}\left(\sqrt{-d}\right)$ is greater than 1.

KEYWORDS:

• Bianchi group
• link complement
• congruence subgroup

2000 AMS SUBJECT CLASSIFICATION:

• Primary 57M25
• Secondary 20H10
• 57M50

Acknowledgments

The authors are very grateful to A. Page who provided presentations for some of the Bianchi groups that were needed in our work. This work was done during visits to the University of Texas by M. D. Baker, and to Université de Rennes 1 by A. W. Reid. A. W. Reid also wishes to thank Max Planck Institüt, Bonn, for its hospitality while this work was completed. The authors are also most grateful to the referee for their careful reading of the article and useful comments that helped improve the organization of the article.

Funding

A. W. Reid is supported in part by the National Science Foundation.