The quasi-positivity and sharp HOMFLY bound conjecture for fillable knots

Let $K$ be a smooth knot type. The HOMFLY bound on the maximum Thurston–Bennequin number of $K$ is the upper bound obtained from the HOMFLY polynomial, and $K$ is fillable if it admits a Lagrangian filling. Fillability conjecture. A smooth knot type $K$ is fillable if and only if it is quasi-positive and the HOMFLY bound on the maximum Thurston–Bennequin number of $K$ is sharp.

This conjecture seeks a necessary and sufficient characterization of Lagrangian fillability using the known necessary condition of quasi-positivity together with sharpness of the HOMFLY bound. The paper establishes that all positive knots are fillable and that strong quasi-positivity alone does not characterize fillability, but the proposed equivalence remains open.