Flat fully augmented links are determined by their complements

Millichap, Christian; Trapp, Rolland

doi:10.2140/agt.2025.25.4839

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ISSN (electronic): 1472-2739

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Flat fully augmented links are determined by their complements

Christian Millichap and Rolland Trapp

Algebraic & Geometric Topology 25 (2025) 4839–4896

DOI: 10.2140/agt.2025.25.4839

Abstract

We show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $𝕊^{3}$ . This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link complements and behave under homeomorphism. One consequence of this analysis is a complete classification of flat fully augmented link complements that admit multiple reflection surfaces. In addition, our work classifies those symmetries of flat fully augmented link complements which are not induced by symmetries of the corresponding link.

Keywords

fully augmented links, hyperbolic links, symmetries of links, hyperbolic 3-manifolds, links determined by their complements

Mathematical Subject Classification

Primary: 57K10, 57K32, 57M50

References

Publication

Received: 18 December 2023

Revised: 31 July 2024

Accepted: 16 September 2024

Published: 20 November 2025

Authors

Christian Millichap
Department of Mathematics Furman University Greenville, SC United States

Rolland Trapp
Department of Mathematics California State University, San Bernardino San Bernardino, CA United States

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Volume 25, issue 8 (2025)