Knots in lattice homology

Assume that F_v0 is a tree with vertex set Vert (F_υ0) = {v₀, v₁, ... , v_n}, and with an integral framing (weight) attached to each vertex except v₀. Assume furthermore that the intersection matrix of G = F_v0 - {v₀} is negative definite. We define a filtration on the chain complex computing the lattice homology of G and show how to use this information in computing lattice homology groups of a negative definite graph we get by attaching some framing to υ₀. As a simple application we produce new families of graphs which have arbitrarily many bad vertices for which the lattice homology groups are isomorphic to the corresponding Heegaard Floer homology groups.