Knots in lattice homology

Abstract

Assume that \Gamma _{v_0} is a tree with vertex set \mathrm{Vert}(\Gamma _{v_0})=\{ v_0, v_1, \ldots , 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 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.