The contact cut graph and a Weinstein $\mathcal{L}$-invariant

TL;DR

This paper introduces the contact cut graph as a new tool in contact geometry, establishing a Weinstein $ ext{L}$-invariant for Weinstein domains, and explores its properties and applications in symplectic topology.

Contribution

It defines the contact cut graph and the Weinstein $ ext{L}$-invariant, linking contact topology with Lefschetz fibrations and multisection diagrams, and provides new insights into Weinstein domain invariants.

Findings

Contact cut graph generalizes Hatcher and Thurston's cut graph for contact geometry.

Weinstein $ ext{L}$-invariant is a new symplectic invariant of Weinstein domains.

Examples with arbitrarily large $ ext{L}$-invariant demonstrate the invariant's variability.

Abstract

We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $\mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $\mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $\mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $\mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $\mathcal{L}$-invariant.