Grid diagrams and Heegaard Floer invariants

TL;DR

This paper provides combinatorial methods to compute Heegaard Floer homology and four-manifold invariants using grid diagrams, simplifying calculations for arbitrary three- and four-manifolds.

Contribution

It introduces a new combinatorial framework based on grid diagrams for calculating Heegaard Floer invariants of three- and four-manifolds.

Findings

Combinatorial descriptions of Heegaard Floer homology for any three-manifold.

Explicit combinatorial formulas for mod 2 Ozsvath-Szabo invariants of four-manifolds.

Simplification of complex topological invariants through grid diagram techniques.

Abstract

We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams.