Identify a DT/stability interpretation for the skein‑valued pentagon identity

Identify or develop a categorical or Donaldson–Thomas/stability‑conditions interpretation for the skein‑valued pentagon identity in the HOMFLYPT skein (e.g., on the punctured torus) analogous to the known derivations of the quantum pentagon identity via wall‑crossing in stability spaces and DT invariants.

Background

From a family of quadratic differentials on a Riemann surface, the authors derive a skein‑valued version of the pentagon identity using their wall‑crossing framework and skein dilogarithms. For the usual quantum pentagon identity, there are well‑known derivations via wall‑crossing in spaces of stability conditions and interpretations in terms of Donaldson–Thomas invariants of CY3 categories.

However, the authors explicitly state that they do not know a similar interpretation for the skein‑valued pentagon identity. Clarifying such an interpretation would connect skein‑valued enumerative geometry to DT theory or stability conditions and could illuminate deeper categorical or representation‑theoretic structures underlying skein dilogarithms.