Exact Lagrangian fillability of 3-braid closures

TL;DR

This paper characterizes when 3-braid closures in contact 3-space admit exact Lagrangian fillings, linking fillability to quasipositivity and the sharpness of the HOMFLY bound, and constructs explicit Legendrian representatives.

Contribution

It provides a complete characterization of exact Lagrangian fillability for 3-braid closures, supporting the orientable fillability conjecture and explicitly constructing Legendrian representatives.

Findings

Orientably fillable 3-braid closures are exactly the quasipositive ones with sharp HOMFLY bound.

Constructed explicit Legendrian representatives with maximum Thurston-Bennequin number.

Supported the conjecture relating fillability to quasipositivity and HOMFLY bound sharpness.

Abstract

We determine when a Legendrian quasipositive 3-braid closure in standard contact $\mathbb{R}^3$ admits an orientable or non-orientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of Hayden and Sabloff, showing that a 3-braid closure is orientably exact Lagrangian fillable if and only if it is quasipositive and the HOMFLY bound on its maximum Thurston-Bennequin number is sharp. Of possible independent interest, we construct explicit Legendrian representatives of quasipositive 3-braid closures with maximum Thurston-Bennequin number.