Conjectured surjection Sk(T−D) → E_{q,t}

Establish a surjective algebra homomorphism from the HOMFLYPT skein algebra Sk(T−D) of the once‑punctured torus to the elliptic Hall algebra E_{q,t} by constructing a map Sk(T−D) → E_{q,t} and proving that it is surjective on the full skein algebra (beyond the currently known maps defined only on subalgebras such as Sk^+(T−D)).

Background

The paper recalls a conjecture from Morton–Samuelson (MS21) asserting the existence of a surjective homomorphism from the full skein algebra of the once‑punctured torus Sk(T−D) to the elliptic Hall algebra E_{q,t}.

While the present work (combining MS21 and BCMN23) establishes such a map on the positive subalgebra Sk^+(T−D) → E^+_{q,t} (and notes further extensions in GL23 to a larger subalgebra), the surjectivity for the full algebra Sk(T−D) remains conjectural. This conjecture is significant in relating skein-theoretic structures with representation‑theoretic objects like the elliptic Hall algebra.