Deformations of lattice cohomology and the upsilon invariant

TL;DR

This paper introduces deformations of lattice cohomology linked to knot homologies, establishing their equivalence with analytic theories and providing combinatorial formulas for the upsilon invariant.

Contribution

It presents new deformations of lattice cohomology related to knot homologies and proves their equivalence with analytic theories for many knots.

Findings

Established equivalence between deformed lattice cohomology and analytic knot theory

Derived combinatorial formulas for the upsilon invariant

Extended the understanding of knot invariants through cohomological deformations

Abstract

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.