Knots in lattice homology

Abstract

Assume that Gamma(v0) is a tree with vertex set Vert(Gamma(v0)) = 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(v0) - 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 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.