We prove the full Fock-Goncharov conjecture for Aₒ₋䃒, _₆, --the A-cluster variety associated to representation of SL₂ local systems on most punctured surfaces with at least 2 punctures in the classical q 1 setting, that is, the tagged skein algebra coincides with the upper cluster algebra (namely Sk^ta=U () or mid (A) =up (A) ), with methods being potentially useful to tackle the quantum case. We deduce similar results for the Roger Yang skein algebra via a birational geometric description, obtaining Sk^RY=U () vᵢ^1 as conjectured by Shen, Sun and Weng, proving important algebraic properties of Sk^RY including normality and Cohen-Macaulayness. Our result is complementary to what and Mandel and Qin have shown in arXiv: 2301. 11101 for surface with marked points, based on arXiv: 1411. 1394. The once-punctured case where cluster structures are significantly different is also discussed in the paper, and relevant conjectures are proposed (and proved in the once-punctured torus case). By contrast, we define the ordinary cluster algebra with potentials added A () vᵢ^1, introduced by Shen, Sun and Weng, which is shown to be usually smaller than Sk^RY and U () vᵢ^1. This strengthen the result of arXiv: 2201. 08833 that the classical A=U fails for ₆, with g 1, p 1.
Enhan Li (Wed,) studied this question.