Sken and cluster algebras of punctured surfaces

We prove the full Fock-Goncharov conjecture for A_SL₂,Σg,p--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 Skᵗᵃ=U(Σ) 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 SkRY=U(Σ)[vᵢ±1] as conjectured by Shen, Sun and Weng, proving important algebraic properties of SkRY 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(Σ)[vᵢ±1], introduced by Shen, Sun and Weng, which is shown to be usually smaller than SkRY and U(Σ)[vᵢ±1]. This strengthen the result of arXiv:2201.08833 that the classical $A=U$ fails for Σg,p with g≥ 1, p≥ 1.