All principal congruence link groups

TL;DR

This paper systematically enumerates all principal congruence link complements in the 3-sphere, addressing a longstanding question posed by Thurston and expanding the understanding of these special hyperbolic 3-manifolds.

Contribution

It provides a complete classification of principal congruence link groups in $S^3$, filling a gap in the topology of hyperbolic 3-manifolds.

Findings

All principal congruence link complements in $S^3$ are enumerated.

The classification confirms the finiteness of such link groups.

Answers a question posed by Thurston regarding these link complements.

Abstract

We enumerate all the principal congruence link complements in $S^3$, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426).