Knots in lattice homology

Abstract

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