Spectral order for contact manifolds with convex boundary

Abstract

We extend the Heegaard Floer homological definition of spectral order for closed contact $3$–manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact $3$–manifolds with convex boundary. We show that the order of a codimension-zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a Giroux $2\pi$–torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).