arXiv research

Decomposes $SL_3$ skein algebras for surfaces.

Survey of stated skein modules/algebras of 3-manifolds/surfaces.

Quantum cluster algebras for surfaces with coefficients defined using skein theory.

Study of cluster and skein algebras for surfaces, showing their connection.

Method computes centers of Poisson and skein algebras for loops on surfaces.

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…

Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.

New method computes automorphisms of surface groups using skein algebras.

We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rel…

In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary compone…

Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.

By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives …

Quantized Coulomb branches linked to skein algebras.

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…

Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.

Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.

Study resolves conjecture linking two algebraic structures on surfaces.

Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…

Study proves certain algebraic structures are symmetric Frobenius algebras.

Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.

Compute central extension of mapping class group from stated skein algebra

Quantum traces embed into quantum tori for surface skein algebras.

Quantum duality map extended to general marked surfaces and its compatibility with skein algebras proven.

Researchers prove positivity of skein algebra structure constants for specific surfaces.

Developed a theory for $SU_3$-skein algebras on surfaces, proving finitely generated property.

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 ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a topological interpretation for its structure morphisms. We also show that its sta…

This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine $k$-algebra over an algebraically cl…

We give an explicit formula for the action of the Dehn twist along a simple closed curve in a compact connected oriented surface on the completion of the filtered skein modules. To do this, we introduce filtrations of the Kauffman bracket skein algebra and the Kauffman bracket skein modules on the surface.

Let $Σ$ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the skein algebra of $Σ$ over the field of rational functions can be algebraically generated by a finite number of simple closed curves that are naturally associated to certain generators of the…

The paper presents a new algebraic structure for planar surfaces.

We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.

Skein algebra action is faithful if quantum parameter isn't a root of 1.

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that…

This paper is focused on the structure of the Kauffman bracket skein algebra of a punctured surface at roots of unity. A criterion that determines when a collection of skeins forms a basis of the skein algebra as an extension over the $SL(2,{\mathbb C})$ characters of the fundamental group of the surface, with appropri…

Presented an algebra structure for a specific geometric surface.

Study $G_2$ skein algebra elements using Kuperberg webs and threading operations.

Study centers of generalized skein algebras, showing almost Azumaya properties.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

Relates two types of skein algebras using explicit correspondences.

Kauffman bracket skein algebra structure on surfaces defined.

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 an…

We show that the twisted SL(2) skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers.

Monoidal categorifies genus zero skein algebra using K-theory.

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructe…

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 …

Quantum cluster algebra constructed from web skein relations on surfaces.

The paper connects two skein algebras and characterizes their representations.