On the contact class in Heegaard Floer homology

TL;DR

This paper offers a new perspective on the Ozsvath-Szabo contact class in Heegaard Floer homology and proves a characterization of tightness for contact structures with a once-punctured torus open book.

Contribution

It provides an alternate description of the contact class and establishes a criterion linking monodromy properties to tightness in specific contact structures.

Findings

The contact class can be described differently using the authors' approach.

Monodromy is right-veering if and only if the contact structure is tight for the given open book.

The result applies specifically to open books with a once-punctured torus page.

Abstract

We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,\xi) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.