Tight contact structures on laminar free hyperbolic three-manifolds

TL;DR

This paper constructs tight contact structures on certain hyperbolic 3-manifolds, including those without taut foliations, using contact surgery diagrams and Heegaard Floer homology invariants.

Contribution

It provides the first examples of tight contact structures on hyperbolic 3-manifolds lacking taut foliations, expanding understanding of contact topology in hyperbolic geometry.

Findings

Constructed tight contact structures on specific hyperbolic 3-manifolds.

Used contact surgery diagrams to describe the structures.

Proved tightness via Heegaard Floer contact invariants.

Abstract

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Shareshian, and Stein, and infinitely many of them do not even admit essential laminations as shown by Fenley. In this paper, we construct tight contact structures on a family of 3-manifolds including these examples. These contact structures are described by contact surgery diagrams and their tightness is proved using the contact invariant in Heegaard Floer homology.