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Relation between H_kappa and a derived HOMFLYPT skein algebra of S^2

Determine the precise mathematical relationship between the differential graded algebra H_kappa—defined in the paper as a unital differential graded algebra over Z[ℏ] generated by braid generators T_1, …, T_{κ−1} satisfying Hecke relations and an additional generator x_1 with specified differential and commutation relations—and a rigorously formulated derived HOMFLYPT skein algebra of the 2-sphere S^2. Develop a precise definition of the derived HOMFLYPT skein algebra for S^2 and ascertain whether H_kappa is quasi-isomorphic (or otherwise canonically equivalent) to this derived skein algebra.

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Background

The authors construct a Morse-type based multiloop A_infinity-algebra and show it is equivalent to the wrapped higher-dimensional Heegaard Floer algebra. For closed surfaces other than S2, this recovers the braid skein algebra, but for S2 the resulting algebra is not concentrated in degree zero, leading them to identify a new differential graded algebra H_kappa.

They prove that the based multiloop A_infinity-algebra for S2 is quasi-equivalent to H_kappa, and suggest H_kappa has a natural quantum-topological interpretation of derived nature. This motivates defining an appropriate notion of a derived HOMFLYPT skein algebra for surfaces and comparing it precisely to H_kappa.

References

Comparing Theorem~\ref{thm: multiloop algebra for S2 is Hn} and Corollary~\ref{cor: equivalent to braid skein algebra} suggests that the DGA $H_\kappa$ should admit a natural interpretation in terms of quantum topology. As it is of derived nature, it is necessary to develop a notion of a derived HOMFLYPT skein algebra of a surface. The natural guess here is that the approach of factorization homology developed in should provide the necessary tool for such computations. Therefore we raise the following:

Question What is the precise relation between the DGA $H_\kappa$ and the appropriate notion of a derived HOMFLYPT skein algebra of $S2$?

— Morse theory of loop spaces and Hecke algebras (2503.07543 - Honda et al., 10 Mar 2025) in Introduction (Section 1)