Characterizing Dehn surgeries on links via trisections

TL;DR

This paper explores the relationship between Dehn surgeries on links and trisections of 4-manifolds, introduces potential counterexamples to a conjecture, and provides tools to analyze trisection reducibility.

Contribution

It establishes new connections between Dehn surgery and trisections, and introduces an analog of the Casson-Gordon Rectangle Condition for trisections.

Findings

Counterexamples to the Generalized Property R Conjecture imply genus four trisections of the 4-sphere.

An analog of the Casson-Gordon Rectangle Condition for trisections is developed.

Potential non-standard genus four trisections of the 4-sphere are identified.

Abstract

We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard. Finally, we give an analog of the Casson- Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection.