An official website of the United States government
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 .
more »
« less
This content will become publicly available on August 1, 2026
Miraculous Cancellations and the Quantum Frobenius for SL 3 Skein Modules
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 $$\mathcal{O}_{q}(SL_{3}).$$
more »
« less
- Award ID(s):
- 2136090
- PAR ID:
- 10627379
- Publisher / Repository:
- Oxford Academic
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 15
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.more » « less
-
Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $$\text{Sym}(n)$$ (in the sofic case) or the finite-dimensional unitary groups $$\text{U}(n)$$ (in the hyperlinear case)? In the case of $$\text{U}(n)$$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $$\text{U}(n)$$ with respect to the Frobenius norm $$\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability , that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $$p$$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.more » « less
-
The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $$3$$-manifolds are finitely generated over $$\Q(A)$$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $$S(M,\Q[A^\pmo])$$ of $$M$$ is tame (e.g. finitely generated over $$\Q[A^{\pm 1}]$$), and the $$SL(2, \C)$$-character scheme is reduced, then the dimension $$\dim_{\Q(A)}\, S(M, \Q(A))$$ is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating $$\dim_{\Q(A)}\, S(M, \Q(A))$$ to the Abouzaid-Manolescu $$SL(2,\C)$$-Floer theoretic invariants, for infinite families of 3-manifolds. We prove a criterion for reducedness of character varieties of closed $$3$$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $$1$$ over $$\Q(A)$$.more » « less
-
Abstract Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov–Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over {S^{2}}.more » « less
