On the contact Ozsvath-Szabo invariant

TL;DR

This paper demonstrates that the contact Ozsvath-Szabo invariant and the hat version of Heegaard Floer homology can be computed combinatorially from contact surgery diagrams, simplifying calculations in contact topology.

Contribution

It introduces a method to compute the contact Ozsvath-Szabo invariant combinatorially from contact surgery diagrams, building on previous algorithms and descriptions.

Findings

Combinatorial calculation of the contact Ozsvath-Szabo invariant is possible from surgery diagrams.

Provides detailed examples and shortcuts for computations.

Extends the algorithmic approach to contact topology invariants.

Abstract

Sarkar and Wang proved that the hat version of Heegaard Floer homology group of a closed oriented 3-manifold is combinatorial starting from an arbitrary nice Heegaard diagram and in fact every closed oriented 3-manifold admits such a Heegaard diagram. Plamenevskaya showed that the contact Ozsvath-Szabo invariant is combinatorial once we are given an open book decomposition compatible with a contact structure. The idea is to combine the algorithm of Sarkar and Wang with the recent description of the contact Ozsvath-Szabo invariant due to Honda, Kazez and Matic. Here we simply observe that the hat version of the Heegaard Floer homology group and the contact Ozsvath-Szabo invariant in this group can be combinatorially calculated starting from a contact surgery diagram. We give detailed examples pointing out to some shortcuts in the computations.