Trisections and link surgeries

TL;DR

This paper explores the relationship between link surgeries and trisection decompositions of 4-manifolds, identifying specific link families with particular surgery properties and providing evidence towards a broader conjecture.

Contribution

It describes families of links on Heegaard surfaces with surgeries yielding connected sums of S^1×S^2, and offers partial evidence regarding the classification of such links.

Findings

Identified link families with surgeries producing connected sums of S^1×S^2

Provided evidence suggesting not all such links belong to these families

Analyzed the implications for 4-manifold trisection theory

Abstract

We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby. These links lie on Heegaard surfaces in $\#^j S^1 \times S^2$ and have surgeries yielding $\#^k S^1 \times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.