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- W4386942683 abstract "When the quantum parameter $q^{1/2}$ is a root of unity of odd order. The stated skein module $S_{q^{1/2}}(M,mathcal{N})$ has an $S_{1}(M,mathcal{N})$-module structure, where $(M,mathcal{N})$ is a marked three manifold. We prove $S_{q^{1/2}}(M,mathcal{N})$ is a finitely generated $S_{1}(M,mathcal{N})$-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 $S_{q^{1/2}}(M,mathcal{N})$ over $S_{1}(M,mathcal{N})$ when $M$ is compact. For a pb surface $Sigma$, we use $S_{q^{1/2}}(Sigma)^{(N)}$ to denote the image of the Frobenius map when $q^{1/2}$ is a root of unity of odd order $N$. Then $S_{q^{1/2}}(Sigma)^{(N)}$ lives in the center of the stated skein algebra $S_{q^{1/2}}(Sigma)$. Let $widetilde{S_{q^{1/2}}(Sigma)^{(N)}}$ be the field of fractions of $S_{q^{1/2}}(Sigma)^{(N)}$, and $widetilde{S_{q^{1/2}}(Sigma)}$ be $S_{q^{1/2}}(Sigma)otimes_{S_{q^{1/2}}(Sigma)^{(N)}} widetilde{S_{q^{1/2}}(Sigma)^{(N)}}$. Then we show the dimension of $widetilde{S_{q^{1/2}}(Sigma)}$ over $widetilde{S_{q^{1/2}}(Sigma)^{(N)}}$ is $N^{3r(Sigma)}$ where $r(Sigma)$ equals to the number of boundary components of $Sigma$ minus the Euler characteristic of $Sigma$." @default.
- W4386942683 created "2023-09-22" @default.
- W4386942683 creator A5056709272 @default.
- W4386942683 date "2023-09-19" @default.
- W4386942683 modified "2023-09-27" @default.
- W4386942683 title "Finiteness and dimension of stated skein modules over Frobenius" @default.
- W4386942683 doi "https://doi.org/10.48550/arxiv.2309.10920" @default.
- W4386942683 hasPublicationYear "2023" @default.
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