What question did this study set out to answer?

This work aims to determine the conditions under which a Legendrian quasipositive 3-braid closure admits an exact Lagrangian filling.

January 23, 2026

Exact Lagrangian fillability of 3‐braid closures

Key Points

This work aims to determine the conditions under which a Legendrian quasipositive 3-braid closure admits an exact Lagrangian filling.
Analyzed Legendrian quasipositive closures in standard contact settings.
Evaluated the sharpness of the HOMFLY bound on Thurston–Bennequin numbers.
Constructed explicit Legendrian representatives of quasipositive 3-braid closures.
Established that orientable exact Lagrangian fillability is linked to quasipositivity and sharp HOMFLY bounds.
Provided explicit examples of Legendrian representatives achieving maximum Thurston–Bennequin numbers.

Abstract

Abstract We determine when a Legendrian quasipositive 3‐braid closure in the standard contact admits an orientable or nonorientable 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.

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