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Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey

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Abstract

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces) and describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.

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Notes

  1. In this paper a \(\partial \mathfrak {S}\)-tangle diagram is a positively ordered \(\partial \mathfrak {S}\)-tangle diagram of [40].

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Acknowledgments

The authors would like to thank F. Bonahon, F. Costantino, C. Frohman, J. Kania-Bartoszynska, T. Schedler, A. Sikora, and M. Yakimov for helpful discussions. The first author is supported in part by NSF grant DMS 1811114.

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  1. School of Mathematics, 686 Cherry Street, Georgia Tech, Atlanta, GA, 30332, USA

    Thang T. Q. Lê & Tao Yu

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  1. Thang T. Q. Lê
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Lê, T.T.Q., Yu, T. Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey. Acta Math Vietnam 46, 265–287 (2021). https://doi.org/10.1007/s40306-021-00417-2

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