Knots in lattice homology

Assume that \Gamma_{v_0} is a tree with vertex set Vert(\Gamma_{v_0})={v_0, v_1,..., v_n}, and with an integral framing (weight) attached to each vertex except v_0. Assume furthermore that the intersection matrix of G=\Gamma_{v_0}-{v_0} 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 v_0. As a simple application we produce families of graphs which have arbitrarily many bad vertices for which the lattice homology groups are shown to be isomorphic to the corresponding Heegaard Floer homology groups.
Comment: 45 pages, 9 figures