arXiv research

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

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

Let $A\neq 0$ be a complex number with $|A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Ku…

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…

We generalize the asymptotic faithfulness of the skein quantum $SU(2)$ representations of mapping class groups of orientable closed surfaces to skein $SU(3)$. Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebra…

Paper compares skein modules to Kauffman bracket modules.

We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…

We study the structure of the Kauffman algebra of a surface with parameter equal to sqrt(-1). We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat connections in a trivial SU(2)-bundle over the surface. We analyse the…

SU(3) instanton homology counts Tait colorings for webs and foams.

Relates two types of skein algebras using explicit correspondences.

Decomposes $SL_3$ skein algebras for surfaces.

Survey on stated skein algebras and their representations.

Center identified in stated skein algebra for quantum traces.

Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in the context of mutant knots are also discussed.

For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link…

Paper presents skein algebras for spheres with punctures.

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

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

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

Survey of stated skein modules/algebras of 3-manifolds/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…

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

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

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 …

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

Finite presentations for skein algebras linked to gauge field theory.

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of…

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

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

The paper connects two skein algebras and characterizes their representations.

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

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

Monoidal categorifies genus zero skein algebra using K-theory.

Study proves certain algebraic structures are symmetric Frobenius algebras.

This thesis explores DAHA representations using stated skein theory.

Proves a pentagon relation in skein theory.

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

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

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…

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

Polynomial algorithm for multiplication on one-hole torus skein algebra.

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

Presented an algebra structure for a specific geometric surface.

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

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…

New method computes automorphisms of surface groups using skein algebras.

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…