Strong Heegaard diagrams and strong L-spaces

TL;DR

This paper investigates strong L-spaces, a class of 3-manifolds with simple Heegaard diagrams, providing evidence they are branched double covers of alternating links and classifying certain cases.

Contribution

It introduces the concept of strong L-spaces, classifies genus two cases explicitly, and shows finiteness results for strong L-spaces with bounded homology order.

Findings

Strong L-spaces admit simple Heegaard diagrams.

Genus two strong L-spaces are classified explicitly.

Finitely many strong L-spaces have bounded first homology order.

Abstract

We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L-space admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong L-spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph theoretic. We discuss many related results and questions.