Knot lattice homology in L-spaces

TL;DR

This paper demonstrates an equivalence between knot lattice homology and knot Floer homology for knots in L-spaces, and shows isomorphisms between lattice and Heegaard Floer homologies for certain negative definite plumbing trees.

Contribution

It establishes a new equivalence between knot lattice homology and knot Floer homology, and proves isomorphisms for specific classes of plumbing graphs.

Findings

Knot lattice homology is equivalent to knot Floer homology in L-spaces.

For certain negative definite plumbing trees, lattice homology is isomorphic to Heegaard Floer homology.

The results unify different invariants in low-dimensional topology.

Abstract

We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)$ is isomorphic to the Heegaard Floer homology HF^-(Y_G) of the corresponding rational homology sphere Y_G.