A spanning tree model for the Heegaard Floer homology of a branched double-cover

TL;DR

This paper introduces a new spanning tree model for computing the Heegaard Floer homology of branched double covers of links, connecting link diagrams to Floer homology computations.

Contribution

It provides a novel combinatorial model relating Kauffman states to Heegaard Floer chain complexes for branched double covers.

Findings

Computed specific examples of (HF) for certain links.

Proposed a conjecture on a delta-grading structure for (HF).

Established a link between link diagrams and Floer homology computations.

Abstract

Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.