Knot lattice homology in L-spaces

Abstract

We show that the knot lattice homology of a knot in an [Formula: see text]-space is chain homotopy equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring [Formula: see text]). Suppose that [Formula: see text] is a negative definite plumbing tree which contains a vertex [Formula: see text] such that [Formula: see text] is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology [Formula: see text] is isomorphic to the Heegaard Floer homology [Formula: see text] of the corresponding rational homology sphere [Formula: see text].