Abstract

Co-authors: Cagatay Kutluhan ( University at Buffalo), Jeremy Van Horn-Morris (University of Arkansas), Andy Wand (University of Glasgow)

In this joint work with Kutluhan, Van Horn-Morris and Wand, we define the {\it spectral order} invariant of contact structures in dimension three by refining the contact invariant from Heegaard Floer homology. This invariant takes values in the set \mathbb{Z}_{\geq0}\cup\{\infty\}. It is zero for overtwisted contact structures, \infty for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than that given by the Heegaard Floer contact invariant, and an obstruction to existence of Stein cobordisms between contact 3-manifolds. We show this by exhibiting an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values.