Flat fully augmented links are determined by their complements

TL;DR

This paper proves that flat fully augmented links are uniquely determined by their complements, providing a classification of their symmetries and reflection surfaces, and analyzing how geometric features behave under homeomorphisms.

Contribution

It establishes that flat fully augmented links are uniquely determined by their complements and classifies their symmetries and reflection surfaces.

Findings

Two flat fully augmented links with homeomorphic complements are equivalent.

Complete classification of flat fully augmented link complements with multiple reflection surfaces.

Identification of symmetries not induced by the link itself.

Abstract

In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $\mathbb{S}^{3}$. 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.