Manifolds with small Heegaard Floer ranks

TL;DR

This paper characterizes certain three-manifolds with minimal Heegaard Floer homology and explores the relationship between Khovanov homology and unknot detection, revealing a unique classification and connections between invariants.

Contribution

It classifies irreducible three-manifolds with positive first Betti number and minimal Heegaard Floer homology, and links Khovanov homology detection properties to these manifolds.

Findings

Only zero-framed surgery on the trefoil has Heegaard Floer rank two among such manifolds.

Classified links whose branched double covers produce this manifold.

Established that Khovanov homology detects the unknot if and only if it detects the two-component unlink.

Abstract

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.