2023/09/19 by Zhihao Wang, Wang, Zhihao
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2309.10920
openalex publication_date 2023/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When the quantum parameter q1/2 is a root of unity of odd order. The stated skein module Sq1/2(M,N) has an S1(M,N)-module structure, where (M,N) is a marked three manifold. We prove Sq1/2(M,N) is a finitely generated S1(M,N)-module when M is compact, which furthermore indicates the reduced stated skein module for the compact marked three manifold is finite dimensional. We also give an upper bound for the dimension of Sq1/2(M,N) over S1(M,N) when M is compact. For a pb surface Σ, we use Sq1/2(Σ)(N) to denote the image of the Frobenius map when q1/2 is a root of unity of odd order N. Then Sq1/2(Σ)(N) lives in the center of the stated skein algebra Sq1/2(Σ). Let \widetildeSq1/2(Σ)(N) be the field of fractions of Sq1/2(Σ)(N), and \widetildeSq1/2(Σ) be Sq1/2(Σ)⊗_Sq1/2(Σ)(N) \widetildeSq1/2(Σ)(N). Then we show the dimension of \widetildeSq1/2(Σ) over \widetildeSq1/2(Σ)(N) is N3r(Σ) where r(Σ) equals to the number of boundary components of Σ minus the Euler characteristic of Σ.