Heegaard Floer homology for manifolds with torus boundary: properties and examples

TL;DR

This paper explores properties of Heegaard Floer homology for manifolds with torus boundary, linking it to knot invariants, Thurston norm, and Turaev torsion, with geometric and symmetry insights and numerous examples.

Contribution

It provides new geometric descriptions, symmetry results, and connections to other theories for Heegaard Floer homology in manifolds with torus boundary.

Findings

Established a symmetry under spin^c conjugation.

Linked Heegaard Floer homology to knot Floer homology and Thurston norm.

Provided geometric descriptions and numerous examples.

Abstract

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under spin$^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and $HF^\pm$. Many examples are included.