# Knots in lattice homology

### Peter S. Ozsváth

Princeton University, USA### András I. Stipsicz

Hungarian Academy of Sciences, Budapest, Hungary### Zoltán Szabó

Princeton University, USA

## Abstract

Assume that $Γ_{v_{0}}$ is a tree with vertex set $Vert(Γ_{v_{0}})={v_{0},v_{1},…,v_{n}}$, and with an integral framing (weight) attached to each vertex except $v_{0}$. Assume furthermore that the intersection matrix of $G=Γ_{v_{0}}−{v_{0}}$ 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.

## Cite this article

Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó, Knots in lattice homology. Comment. Math. Helv. 89 (2014), no. 4, pp. 783–818

DOI 10.4171/CMH/334