Quantum holonomic link invariants derived from stated skein algebras

Abstract: We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant is derived from De Concini-Kac quantum coadjoint action lifted at the level of stated skein algebras. A key feature is the fact that the Drinfeld double of the quantum Borel algebra admits a natural interpretation as the reduced stated skein algebra of a once-punctured bigon from which we deduce a relation between our link invariants and quantum group constructions of Blanchet-Geer-Patureau Mirand-Reshetikhin. Using Bonahon-Wong quantum trace, we also relate our construction to quantum hyperbolic geometry, hence to Kashaev and Baseilhac-Benedetti constructions. We deduce from this relation explicit formulas for the R-matrices, which permit to compute the link invariants explicitly. In particular, we derive an alternative conceptual proof of the Murakami-Murakami relation between the Kashaev invariant and the colored Jones polynomials.