Classification of trisections and the Generalized Property R Conjecture

TL;DR

This paper demonstrates that many unbalanced four-manifold trisections are standard, introduces potential non-standard examples of the four-sphere, and proves a stable version of the Generalized Property R Conjecture for certain links.

Contribution

It establishes standardness for a large class of unbalanced four-manifold trisections and proves a stable Property R Conjecture variant for links with limited tunnel number.

Findings

Most unbalanced four-manifold trisections are standard.

Identifies a family of potentially non-standard four-sphere trisections.

Proves a stable version of the Generalized Property R Conjecture for c-component links with tunnel number at most c.

Abstract

We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisections of the four-sphere. As an application, we prove a stable version of the Generalized Property R Conjecture for $c$-component links with tunnel number at most $c$.