A spectral sequence on lattice homology

TL;DR

This paper constructs a spectral sequence linking lattice homology of plumbing trees to Heegaard Floer homology of 3-manifolds, revealing isomorphisms in cases with limited 'bad' vertices.

Contribution

It introduces a spectral sequence connecting lattice and Heegaard Floer homologies, providing new insights into their relationship for specific graph classes.

Findings

Spectral sequence from lattice to Heegaard Floer homology

Isomorphism for graphs with up to two 'bad' vertices

Enhanced understanding of 3-manifold invariants

Abstract

Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heegaard Floer homology of the underlying 3-manifold.