Dehn surgery, rational open books and knot Floer homology

TL;DR

This paper links rational open book decompositions to contact invariants via knot Floer homology, enabling the detection of tight contact structures on manifolds obtained through surgery, with implications for lens space surgeries.

Contribution

It provides a method to compute Heegaard Floer contact invariants from knot Floer homology of the binding, connecting open book decompositions and contact topology.

Findings

Contact invariants can be computed using knot Floer homology.

Certain surgeries on open book bindings produce tight contact structures.

Applications to lens space surgeries are explored.

Abstract

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgeries are discussed.