Deformations of lattice cohomology and the upsilon invariant

We introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsv' ath, Stipsicz and Szab' o in \cite{OSS4}. By means of holomorphic triangles counting, we prove equivalence with the analytic theory for a wide class of knots. This yields combinatorial formulae for the upsilon invariant.