Bordered contact invariants and half Giroux torsion

TL;DR

This paper constructs examples of contact 3-manifolds with half Giroux torsion that have non-vanishing contact invariants, challenging previous conjectures and exploring the relationship between torsion and fillability.

Contribution

It introduces new counterexamples to Ghiggini's conjecture using bordered contact invariants and analyzes the impact of torsion on contact invariants.

Findings

Existence of infinitely many contact 3-manifolds with non-vanishing invariants despite half Giroux torsion.

Counterexamples to Ghiggini's conjecture regarding torsion and fillability.

Identification of minimal twisting (2π) needed for contact invariant vanishing.

Abstract

We show that there exist infinitely many closed contact 3-manifolds containing half Giroux torsion along a separating torus whose contact invariants do not vanish. This provides counterexamples to Ghiggini's conjecture and suggests that separating half Giroux torsion may not obstruct symplectic fillability. The main tools are the bordered contact invariants recently developed by the authors and the innermost contact structures on knot complements. We also show that there exists a closed contact 3-manifold containing convex torsion along a separating torus with non-vanishing contact invariant, which implies that $2\pi$ is the minimal amount of twisting necessary to ensure vanishing of the contact invariant.