Skein Algebra Research Articles

Mapping Class Group Representations Derived from Stated Skein Algebras

We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.

Stated skein algebras of surfaces

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$ thus providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin–Turaev functor, and in particular, we recover the dual $R$-matrix for $\mathcal{O}{{q^2}}(\operatorname{SL}(2)$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is a comodule algebra over $\mathcal{O}{{q^2}}(\operatorname{SL}(2))^{\otimes n}$ and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as “gluing on a triangle”; then we recover topologically some bialgebras in the category of $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$-comodules, among which the “transmutation” of $\mathcal{O}\_{{q^2}}(\operatorname{SL}(2))$. We also provide an operadic interpretation of stated skein algebras as an example of a “geometric non-symmetric modular operad”. In the last part of the paper, we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon–Wong's quantum trace map and interpret skein algebras in the classical limit when $q\to 1$ as regular functions over a suitable version of moduli spaces of twisted bundles.

The skein algebra of the Borromean rings complement

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the [Formula: see text]-character variety. In this paper, we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in [Formula: see text].

Relating Stated Skein Algebras and Internal Skein Algebras

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., L\^e T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision properties of stated skein algebras when $\mathcal{V} = \mathcal{U}_{q^2}(\mathfrak{sl}_2)\text{-}{\rm mod}^{\rm fin}$. Our proofs are mostly based on skein theory and we do not require the reader to be familiar with the formalism of higher categories.

Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong’s quantum trace map. The second embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller’s quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension.

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov–Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov–Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong’s quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the [Formula: see text]-character variety. This algebraic morphism shares many resemblances with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative [Formula: see text] character variety.

The action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the 3-twist knot complement

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern–Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, the first author, and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern–Simons theory where examples are scarce.

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

The Kauffman Skein Algebra of the Torus

Abstract We give a presentation of the Kauffman (BMW) skein algebra of the torus. This algebra is the “type $BCD$” analogue of the Homflypt skein algebra of torus, which was computed in earlier work of the 1st and 3rd authors [17]. This suggests the existence of a “type $BCD$” version of the Hall algebra of an elliptic curve [4]. In the appendix, we show this presentation is compatible with the Frohman–Gelca description of the Kauffman bracket (Temperley–Lieb) skein algebra of the torus [12].

The Chebyshev–Frobenius homomorphism for stated skein modules of 3-manifolds

We study the stated skein modules of marked \(3-\)manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of \(3-\)manifolds. We show that there exists a Chebyshev–Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev–Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev–Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of \({\mathcal {O}}_{q^2}(SL(2))\) through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig’s Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev–Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.

On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.

Presentations of the Roger–Yang generalized skein algebra

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.

Faithfullness of Geometric Action of Skein Algebras

We show that the action of the Kauffman bracket skein algebra of a surface Σ on the skein module of the handlebody bounded by Σ is faithful if and only if the quantum parameter is not a root of 1.

On the genus two skein algebra

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

Dimension and Trace of the Kauffman Bracket Skein Algebra

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 .

Legendrian skein algebras and Hall algebras

We compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type Stimes {mathbb {R}}. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus.

SU(3)-skein algebras and webs on surfaces

The \(SU_3\)-skein algebra of a surface F is spanned by isotopy classes of certain framed graphs in \(F\times I\) called 3-webs subject to the skein relations encapsulating relations between \(U_q(sl(3))\)-representations. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced \(SU_3\)-skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of F every reduced 3-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional “crossbars” connecting their edges along the ideal triangulation. We show that such canonical position is unique up to “crossbar moves”. That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to \({\mathcal {A}}\)-varieties of Fock–Goncharov and to \(\text {Loc}_{SL(3)}\)-varieties of Goncharov–Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov–Shen.

DAHAs and Skein Theory

We give a skein-theoretic realization of the $\mathfrak{gl}_n$ double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann.

Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces) and describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.

The Roger–Yang skein algebra and the decorated Teichmüller space

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper, we consider surfaces with punctures which are not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang’s Poisson algebra homomorphism is injective, and the skein algebra has no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.