Lattice homology, formality, and plumbed L-space links

TL;DR

This paper introduces a new link lattice complex for plumbed links, demonstrating its homotopy equivalence to the link Floer complex and providing an algorithm to compute these complexes from the Alexander polynomial.

Contribution

It generalizes existing constructions, proves homotopy equivalence for plumbed links, and offers an algorithm for computing link Floer complexes of L-space links.

Findings

Link lattice complex is homotopy equivalent to link Floer complex for plumbed links.

Link Floer complex of a plumbed L-space link is a free resolution of its homology.

Provides an algorithm to compute link Floer complexes from the Alexander polynomial.

Abstract

We define a link lattice complex for plumbed links, generalizing constructions of Ozsv\'ath, Stipsicz and Szab\'o, and of Gorsky and N\'emethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial.