Establish the SFT–skein isomorphism for cotangent bundles of 3‑manifolds

Prove the existence of an isomorphism Ω: SFT^0_bare(S^*M) → Sk(M) for any 3‑manifold M, where SFT^0_bare(S^*M) denotes the degree‑zero (all‑genus) symplectic field theory of the contact manifold S^*M counting bare holomorphic curves asymptotic to Reeb orbits and recorded by their boundary classes, and Sk(M) is the HOMFLYPT skein of M. Construct this isomorphism by counting holomorphic curves in (T^*M, 0_M) with asymptotics at infinity and boundary measured in the skein of M, thereby rigorously realizing the conjectural correspondence used in the paper’s SFT proof sketch.

Background

To give an alternative proof of the skein trace via SFT stretching, the authors posit an isomorphism between a ‘bare curve’ version of the degree‑zero SFT of the unit cosphere bundle S^*M and the HOMFLYPT skein of M. This isomorphism, denoted Ω, would equate SFT curve counts (asymptotic to Reeb orbits) with skein classes recorded by their boundaries in M.

While the paper proceeds by accepting this identification to outline a second proof, the authors explicitly label the isomorphism as conjectural in this work, indicating it is intended to be established elsewhere. A rigorous construction and proof of Ω would solidify the SFT‑based derivation of the skein trace and related wall‑crossing identities.