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1. Stated SL(n$$n$$)‐skein modules and algebras

   https://doi.org/10.1112/topo.12350  

   Lê, Thang_T Q; Sikora, Adam S (September 2024, Journal of Topology)

   Abstract We develop a theory of stated SL()‐skein modules, , of 3‐manifolds marked with intervals in their boundaries. These skein modules, generalizing stated SL(2)‐modules of the first author, stated SL(3)‐modules of Higgins', and SU(n)‐skein modules of the second author, consist of linear combinations of framed, oriented graphs, called ‐webs, with ends in , considered up to skein relations of the ‐Reshetikhin–Turaev functor on tangles, involving coupons representing the anti‐symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3‐manifold along a disk resulting in a 3‐manifold yields a homomorphism for all . That result allows to analyze the skein modules of 3‐manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces , in whose case, is an algebra, denoted by . One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on . (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary every splitting homomorphism is injective and that is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right ‐comodule structure on , or dually, a left ‐module structure. Furthermore, we show that the skein algebra of surfaces glued along two sides of a triangle is isomorphic with the braided tensor product of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of is equivalent to the category of left modules over for surfaces with . We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and . 

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2. Dimension and Trace of the Kauffman Bracket Skein Algebra

   https://doi.org/10.1090/btran/69  

   Frohman, Charles; Kania-Bartoszynska, Joanna; Lê, Thang (July 2021, Transactions of the American Mathematical Society, Series B)

   Let F F be a finite type surface and ζ \zeta a complex root of unity. The Kauffman bracket skein algebra K ζ ( F ) K_\zeta (F) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of K ζ ( F ) K_\zeta (F) over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of F F . 

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3. Miraculous Cancellations and the Quantum Frobenius for SL 3 Skein Modules

   https://doi.org/10.1093/imrn/rnaf226  

   Higgins, Vijay (August 2025, International Mathematics Research Notices)

   Abstract We construct a quantum Frobenius map for the $$SL_{3}$$ skein module of any oriented 3-manifold specialized at a root of unity, and describe the map by way of threading certain polynomials along links. The homomorphism is a higher rank version of the Chebyshev–Frobenius homomorphism of Bonahon–Wong. The strategy builds on a previous construction of the Frobenius map for $$SL_{3}$$ skein algebras of punctured surfaces, using the Frobenius map of Parshall–Wang for the quantum group $$\mathcal{O}_{q}(SL_{3}).$$ 

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4. Set Partitions, Fermions, and Skein Relations

   https://doi.org/10.1093/imrn/rnac110  

   Kim, Jesse; Rhoades, Brendon (May 2022, International Mathematics Research Notices)

   Abstract Let $$\Theta _n = (\theta _1, \dots , \theta _n)$$ and $$\Xi _n = (\xi _1, \dots , \xi _n)$$ be two lists of $$n$$ variables, and consider the diagonal action of $${{\mathfrak {S}}}_n$$ on the exterior algebra $$\wedge \{ \Theta _n, \Xi _n \}$$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$$FDR_n$$ obtained from $$\wedge \{ \Theta _n, \Xi _n \}$$ by modding out by the ideal generated by the $${{\mathfrak {S}}}_n$$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of $${{\mathfrak {S}}}_n$$ on the vector space with basis given by noncrossing set partitions of $$\{1,\dots ,n\}$$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $$FDR_n$$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $${{\mathfrak {S}}}_n$$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution. 

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5. Quantum Cuntz-Krieger algebras

   https://doi.org/10.1090/btran/88  

   Brannan, Michael; Eifler, Kari; Voigt, Christian; Weber, Moritz (October 2022, Transactions of the American Mathematical Society, Series B)

   Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^\ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ C^\ast -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to K K KK -equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C ∗ C^\ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K KK -theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general. 

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