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

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.