Quantum Duality Maps, Skein Algebras and their Ensemble Compatibility

Abstract We generalize the quantum duality map I ₀ I A of Allegretti–Kim (Adv Math 306: 1164–1208, 2017) for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map I ₗ I X on the dual side based on the quantum bracelets basis (Mandel and Qin in arXiv: 2301. 11101 ; Thurston in Proc Natl Acad Sci USA 111 (27): 9725–9732, 2014). Our construction factors through reduced stated skein algebras, based on the quantum trace maps (Lê in Quantum Topol 9 (3): 591–632, 2018) together with an appropriate way of skein lifting of integral A A -laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.