2. Skein modules and algebra
This section has questions related to Skein modules and Skein algebra-
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-
Remark. For n=2, see Constantino-Le’s "Stated skein algebra of surfaces"
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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.-
Remark. Reference: Kinnear arxiv: 2304.07332 "Skein modules of mapping tori"
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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.-
Remark. Definitions from 1. Cautis-Kamnitzer-Morrison 2. Sikora 3. Jordan et. al. 4. Baseilhac-Roche
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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.