arXiv research

Decomposes $SL_3$ skein algebras for surfaces.

The paper studies properties of stated SL(n)-skein algebras and their centers.

Quantum cluster algebra constructed from web skein relations on surfaces.

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

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…

Study centers of quantum tori and skein algebras for even roots of unity.

Found a basis and presentation for a specific algebra.

New central elements found in a quantum algebra related to knot theory.

Study of skein invariants on tori for various groups and quantum parameters.

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $ρ$ of the skein algebra, which is a character $r_ρ\in \ma…

Quantum Frobenius map for $SL_3$ skein modules constructed and described.

Developed a theory of stated SL(n)-skein modules for 3-manifolds.

This paper calculates the skein algebra of the Borromean rings complement.

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

These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL(2,C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety.

Establishes a connection between skein modules and algebraic sets.

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

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

Researchers calculate dimensions of skein modules for 2-torus mapping tori.

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.

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …

The paper provides coordinates for ${ m SL}_3$-web diagrams on surfaces.

Relates two types of skein algebras using explicit correspondences.

Develops quantum character theory for complex reductive groups.

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…

We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tan…

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

We compute the Kauffman skein module of the complement of torus knots in S^3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2,C)-characters tensored with the ring of Laurent polynomials.

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…

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

Diagrammatic calculus proves Alexander polynomial formulas.

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

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

The paper defines a new algebra structure on skein modules of 3-manifolds and shows it's isomorphic to a known algebra.

In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter $q$ is a root of unity. The current paper is devoted to giving a more repr…

Quantum traces map skein algebras to Fock-Goncharov spaces.

New sl(2) action defined on a mathematical module.

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combini…

Quantum traces embed into quantum tori for surface skein algebras.

Cluster algebras match for specific Lie algebras and surfaces.

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms …

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters $χ(F)$ of the fundamental group of $F$. We localize by inverting the nonzero charac…

The paper finds a special pants decomposition for certain surface group representations.

We prove a uniqueness result for finite-dimensional representations of the Kauffman skein algebra $\mathcal{S}_A(S)$ of a surface $S$, when $A$ is a root of unity and when the surface $S$ is a sphere with at most four punctures or a torus with at most one puncture. We show that, if two irreducible representations of $\…

We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl pro…

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

Study of $\mathfrak{sp}_4$-webs on surfaces, proving cluster algebra structure.

Paper proves zero stability for one-row colored sl₃-Jones polynomials.