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2. Skein modules and algebra

This section has questions related to Skein modules and Skein algebra
    1. Peripheral ideal in Kauffman bracket skein algebra

      Problem 2.05.

      Find non-trivial explicit element in the peripheral ideal in the Kauffman bracket skein algebra of the boundary torus for two bridge knots.
        • Problem 2.1.

          Find other classes of knots for which the peripheral ideal is non trivial. For example, trying to find a knot which is not a 2-bridge knot
            • Indecomposable SL_3 webs

              Problem 2.15.

              For a triangulated punctured surface, find the indecomposable SL_3 webs for the Knutsen-Tao cone
                • State Skein algebra of bigons

                  Problem 2.2.

                  Find skeins representing the basis for the isomorphism "Stated Skein Algebra of bigon \cong O_q(SL_n) for n\geq 3" where O_q(SL_n) has Kashiwara-Lusztig canonical basis
                    1. Remark. For n=2, see Constantino-Le’s "Stated skein algebra of surfaces"
                        • KBSM of connected sum of Lens space

                          Problem 2.25.

                          Compute the Kauffman Bracket Skein Module of # Lens space (Haken manifold of finite type) over \mathbb{Z} [A^{\pm}]
                            • SL_n skein modules

                              Problem 2.3.

                              Explore SL_n skein modules for n\geq 3. For example, the dimension < \infty is proved by Gunningham-Jordan-Safranov.
                                1. Remark. Reference: Kinnear arxiv: 2304.07332 "Skein modules of mapping tori"
                                    • SL_n skein modules definition equivalence

                                      Problem 2.35.

                                      Compare the definitions of SL_n skein modules/algebra for either a generic q or q being a root of unity.
                                        1. Remark. Definitions from 1. Cautis-Kamnitzer-Morrison 2. Sikora 3. Jordan et. al. 4. Baseilhac-Roche
                                            • Quantum trace for closed surfaces

                                              Problem 2.4.

                                              For Bloomquist-Karuo-Le quantum trace for closed surface, what happens when we change the pants decomposition? (Hatcher-Thurston move)
                                                • Skein module for 3 manifold with S(\Sigma) action

                                                  Problem 2.45.

                                                  Let S(\Sigma) act on N be a module of the Kauffman Bracket Skein Algebra. Can we determine if N is the skein module of a 3 manifold M with S(\Sigma) action given by \Sigma: \partial M \hookrightarrow M
                                                    • Gluing Stated skein algebra

                                                      Problem 2.5.

                                                      Is there a formula for gluing stated skein algebra along circles?

                                                          Cite this as: AimPL: Quantum invariants and low-dimensional topology, available at http://aimpl.org/quantumlowdimtop.