The Gompf $\theta$-Invariant of Canonical Contact Structures via Legendrian Surgery

TL;DR

This paper provides an explicit Legendrian surgery description of the canonical contact structure on certain plumbed 3-manifolds, derives a formula for Gompf's theta-invariant, and applies it to classify fillability properties.

Contribution

It introduces a closed-form formula for Gompf's theta-invariant for canonical contact structures on plumbed 3-manifolds, extending previous results and providing new applications.

Findings

Explicit Legendrian surgery description of the canonical contact structure.

Closed-form formula for Gompf's theta-invariant in the Seifert fibered case.

Demonstrates minimality of theta-invariant and rules out certain fillings.

Abstract

Let $\Gamma$ be a minimal connected negative-definite plumbing tree with all vertices of genus zero, and let $Y_\Gamma$ be the oriented link of the corresponding normal complex surface singularity, equipped with its canonical contact structure $\xi_{\rm can}$. We give an explicit Legendrian surgery description of $\xi_{\rm can}$, showing that it is the unique consistent diagram-realizable contact structure on $Y_\Gamma$, up to isomorphism. We then derive a closed-form formula for Gompf's $\theta$-invariant of $\xi_{\rm can}$ in the Seifert fibered case, expressed purely in terms of the Hirzebruch--Jung continued fraction expansions of the normalized Seifert invariants, and prove a recursive leaf-to-root formula for arbitrary plumbing trees. The Seifert formula recovers previously known formulas for lens spaces, dihedral manifolds, and small Seifert fibered spaces with complementary legs, and agrees with the N\'emethi--Nicolaescu expression via the classical Hirzebruch--Zagier identity. As a final application we show that $\xi_{\rm can}$ strictly minimizes $\theta$ among all diagram-realizable contact structures on $Y_\Gamma$, and we use this to rule out symplectic rational homology ball fillings for a large class of Stein fillable contact rational homology $3$-spheres.