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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


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      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.


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          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


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              Problem 2.15.

              For a triangulated punctured surface, find the indecomposable $SL_3$ webs for the Knutsen-Tao cone

                • State Skein algebra of bigons


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                  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


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                          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


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                              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


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                                      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


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                                              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


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                                                  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


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                                                      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.