We extend the Heegaard Floer homological definition of spectral order for closed contact
–manifolds
due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact
–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
–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).
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