On Heegaard Floer minimal knots in sutured manifolds

TL;DR

This paper classifies minimal knots in sutured manifolds using Heegaard Floer homology, providing evidence for the conjectured isomorphism with instanton Floer homology and exploring link detection in $S^1\times S^2$.

Contribution

It offers a Heegaard Floer homology classification of minimal knots in sutured manifolds, aligning with prior instanton Floer results, and investigates link detection properties.

Findings

Heegaard Floer classification matches instanton Floer classification where applicable.

Link Floer homology detects spherical braid closures in $S^1\times S^2$.

Supports the conjecture of isomorphism between instanton and Heegaard Floer homologies.

Abstract

Li-Xie-Zhang classified instanton Floer minimal knots in balanced sutured manifolds subject to a condition on the fundamental group. In this paper, we give a similar classification in the Heegaard Floer homology setting. Since our classifications agree when they are both applicable, this provides further evidence for the conjecture of Kronheimer-Mrowka that instanton Floer homology and Heegaard Floer homology are isomorphic. We also study link Floer homology botany question in $S^1\times S^2$, showing that link Floer homology detects spherical braid closures among homologically nontrivial links.