arXiv research

Survey on stated skein algebras and their representations.

The paper extends asymptotic faithfulness to skein quantum representations of mapping class groups.

New invariant derived from skein algebra representations at roots of unity.

This thesis explores DAHA representations using stated skein theory.

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

Study on skein module dimensions at irreducible representations.

Quantum theory constructs a group and skein module for knot complements.

The paper connects two skein algebras and characterizes their representations.

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

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

Constructs quantum representations of mapping class groups using skein theory.

Defines a new 2+1-G-HQFT using graded skein modules.

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…

New relations link knot theory to quiver representations in 3d physics.

New representations of pure braid groups defined from $A_2$ web spaces.

The paper proves unicity for Kauffman bracket skein algebras.

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

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arX…

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 …

Homflypt skein theory and string topology linked via 2-groupoids.

Propose a model-independent axiomatic framework for derived skein theory.

Categorifies weight-preserving maps in sl(2) representations.

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…

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

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

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

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a …

New Langlands duality conjectures for 3-manifold skein modules.

Compute central extension of mapping class group from stated skein algebra

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 $\…

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

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 …

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

DAHA and skein algebra on a double-torus knot are studied.

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces wi…

Establishes a connection between skein modules and algebraic sets.

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

Diagrammatic calculus proves Alexander polynomial formulas.

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

Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton's conjectured inequality and a condition to decide that some polynomials, like the one of 9_{49}, are not adm…

Quantum link invariants derived from skein algebras.

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

Spider category comparison proves equivalence to Sikora's quotient category.

A mathematical isomorphism connects Floer homology to DAHA representations.

New invariant for singular links via bt-algebra.

Quantum $SL_2$ reveals surprising cancellations.

The paper explores basic properties of knot skein invariants.

We give an irreducible decomposition of the so-called local representations (see arXiv:0707.2151) of the quantum Teichmüller space $\mathcal{T}_q(Σ)$ where $Σ$ is a punctured surface of genus $g>0$ and $q$ is a primitive $N$-th root of unity with $N$ odd. As an application, we construct a family of representations of t…