Bridge position of 3-manifolds embedded in the 5-sphere

TL;DR

This paper extends the concept of bridge decompositions to 3-manifolds embedded in the 5-sphere, generalizing classical and recent notions, and provides explicit examples and a unifying technical framework.

Contribution

It introduces bridge decompositions for 3-manifolds in the 5-sphere, proving their existence and encoding embeddings via trivial tangle diagrams.

Findings

Every embedded 3-manifold admits a bridge decomposition.

Embedded 3-manifolds can be represented by four trivial tangle diagrams.

Includes explicit examples like $S^2$-spun knots and ribbon 3-knots.

Abstract

We introduce and study bridge decompositions for 3-manifolds embedded in the 5-sphere. These generalize both the classical notion of bridge position for knots in the 3-sphere and the bridge trisections of surfaces in the 4-sphere due to Meier and Zupan. Our main technical tool is the multisections of 5-manifolds introduced by Aribi, Courte, Golla, and Moussard. We prove that every embedded 3-manifold admits such a decomposition; in particular, any such embedding is encoded by four trivial tangle diagrams. We also present a range of explicit examples, including $S^2$-spun knots and ribbon 3-knots.