Center identified in stated skein algebra for quantum traces.
problem Understanding the center of the stated skein algebra.
method Analyzing the algebra as a generalization of Kauffman bracket skein algebra, focusing on the case when the quantum parameter is a root of unity.
result Simple description and dimension calculation of the center over the center module.
Quantum cluster algebras for surfaces with coefficients defined using skein theory.
problem Defining quantum cluster algebras for surfaces with coefficients.
method Introducing a skein algebra and proving it has a quantum cluster structure.
result The skein algebra of a walled surface naturally generalizes quantum cluster algebras of marked surfaces.
Skein algebra action is faithful if quantum parameter isn't a root of 1.
problem Faithfulness of geometric action of skein algebras.
method Examined the Kauffman bracket skein algebra and its action on handlebody modules.
result The action is faithful if and only if the quantum parameter is not a root of 1.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.
problem Studying representations of Kauffman bracket skein algebras at roots of unity.
method Using the action of the skein algebra on the skein module of the handlebody.
result Explicit reconstruction of unique representation with fixed classical shadow.
Study centers of quantum tori and skein algebras for even roots of unity.
problem Understanding the center of quantum tori and skein algebras for even roots of unity.
method Analyzing quantum tori and skein algebras, computing PI-degree, and decomposing matrices.
result PI-degrees of quantum tori and skein algebras are the same.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
Survey of stated skein modules/algebras of 3-manifolds/surfaces.
problem Understanding stated skein modules/algebras of 3-manifolds/surfaces.
method Discussion of splitting homomorphism, general structures, Frobenius homomorphism, center, dimension, representation theory.
result Skein algebra of non-closed marked surface at any root of 1 is a maximal order.
Quantum duality map extended to general marked surfaces and its compatibility with skein algebras proven.
problem Generalizing quantum duality map to general marked surfaces and proving its compatibility with skein algebras.
method Generalized quantum duality map, reduced stated skein algebras, quantum trace maps, skein lifting.
result Compatibility of quantum duality map with skein algebras proven.
Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.
problem Generalizing skein algebras for surfaces with arbitrary ground rings.
method Constructs LRY skein algebras, quantum traces, and Dehn-Thurston coordinates.
result LRY skein algebras are domains, have degenerations to monomial subalgebras of quantum tori, and are orderly finitely generated.
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…
Finite presentations for skein algebras linked to gauge field theory.
problem Understanding finite presentations for skein algebras and their relationship to gauge field theory.
method Provided finite presentations and deduced properties of stated skein algebras.
result Stated skein algebras are Koszul and isomorphic to quantum moduli algebras in gauge field theory.
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 …
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 …
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…
Polynomial algorithm for multiplication on one-hole torus skein algebra.
problem Complexity of multiplicative structure in skein algebra.
method Provided a polynomial algorithm for one-hole torus.
result Closed form formulas for multiplication of curves with low crossing number.
New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.
problem Proving the structure of SL(n) skein algebra for a specific surface.
method Constructing a linear basis of explicit SL(n) webs, proving spanning and linear independence.
result SL(n) skein algebra of twice punctured sphere is a commutative polynomial algebra in n-1 generators.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
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…
This thesis explores DAHA representations using stated skein theory.
problem Understanding the representation theory of double affine Hecke algebras.
method Combining stated skein theory with DAHA, focusing on the A1 DAHA. result Constructed a module of Laurent polynomials for the A1 DAHA. We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant …
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
Develops quantum character theory for complex reductive groups.
problem Quantum analogue of conjugation equivariant D-modules. method Schur-Weyl functor and double affine Hecke algebra.
result Computes endomorphism algebras of quantum Hotta-Kashiwara modules.
Quantum Frobenius map for SL3 skein modules constructed and described.
problem Constructing a quantum Frobenius map for SL3 skein modules. method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group Oq(SL3). result Described the quantum Frobenius map for SL3 skein modules. Study centers of generalized skein algebras, showing almost Azumaya properties.
problem Understanding the center of generalized skein algebras.
method Generalized skein algebra generated by loops and arcs, computed center, discussed implications.
result Center of Muller-Roger-Yang skein algebra is almost Azumaya.
Study of skein invariants on tori for various groups and quantum parameters.
problem Analysis of G-skein theory invariants on tori for different groups and parameters. method Combinatorial and algebraic methods, including DAHA and skein relations.
result Isomorphisms and homomorphisms between skein algebras and DAHA, proving equivalence of tangles.
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 …
Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.
problem Positivity conjecture for Roger--Yang skein algebras.
method Used explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases.
result Polynomials form a lower bound in the sense of [Lê18] and [LTY21].
Quantum traces map skein algebras to Fock-Goncharov spaces.
problem Establishing quantum traces between skein algebras and Fock-Goncharov spaces.
method Defining and proving properties of quantum traces for SLn-skein algebras. result Existence and properties of quantum traces for SLn-skein algebras. Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of SLd-skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in SLd-skein algebra. Study proves certain algebraic structures are symmetric Frobenius algebras.
problem Understanding algebraic structures in bordered surfaces.
method Analyzing stated skein algebras and their fraction rings.
result Fraction ring of stated skein algebra is a symmetric Frobenius algebra.
Let F be a finite type surface and ζ a complex root of unity. The Kauffman bracket skein algebra Kζ(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 Fie…
A new quantum relation connects exceptional Lie algebras and knots.
problem Understanding the relationship between exceptional Lie algebras and quantum invariants of knots.
method Developed a two-parameter skein relation on trivalent graphs that specializes to exceptional Lie algebras.
result Found a new quantum exceptional polynomial that agrees with classical computations for knots and links.
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 Oq2(SL(2)) providing a topological interpretation for its structure morphisms. We also show that its sta…
Bracelets and theta bases match in various cluster algebras.
problem Matching bracelet and theta bases in cluster algebras.
method Comparing skein and cluster algebras, defining quantum bracelets, and analyzing cluster scattering diagrams.
result Quantum bracelets coincide with theta functions in various cluster algebras.
New basis confirms Thurston's conjecture and reveals knot configurations.
problem Understanding cluster algebras and their bases from surfaces.
method Topological construction of band basis and comparison with Kazhdan-Lusztig type basis.
result Common triangular basis matches band basis in quantum cluster algebras.
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…
Propose a model-independent axiomatic framework for derived skein theory.
problem Derived skein theory of oriented 3-manifolds with coefficients in a ribbon tensor category.
method Design axioms for the 0th homology and gluing.
result Establishes relationships between derived and ordinary skein theory.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…
The Hecke algebra H_n contains well known idempotents E_λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_λ was given by Aiston and Morton. The closure of E_λ becomes an element Q_λ of the skein of the annulus. In this skein, they are known…
Study G2 skein algebra elements using Kuperberg webs and threading operations.
problem Identify and analyze central elements in G2 skein algebra. method Use Kuperberg webs, threading operations, and skein-theoretic arguments.
result Verify central elements and obtain uniqueness of transparent polynomials.
Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.
problem Study sliced skein algebras and their properties.
method Quotient of Kauffman bracket skein algebra, center calculation, PI-degree calculation, fully Azumaya point analysis.
result Center and PI-degree calculations for sliced skein algebras, fully Azumaya points, and simple modules.
Study of sp4-webs on surfaces, proving cluster algebra structure.
problem Understanding sp4-webs and their cluster algebra properties. method Introduced skein algebra and cluster structure, proved positivity.
result Proved Ssp4,ΣZq[∂−1] is a quantum cluster algebra. Developed a theory of stated SL(n)-skein modules for 3-manifolds.
problem Understanding the algebraic structure of 3-manifolds marked with intervals.
method Introduced and studied SL(n)-skein modules and algebras of 3-manifolds, proving homomorphisms and algebra properties.
result Skein algebra of thickened surfaces is isomorphic to Oq(SL(n)) and provides geometric interpretations of quantum group structures. Study the algebraic action of torus on knot complement's skein module.
problem Understand the algebraic structure of knot complements and boundary tori.
method Analyze the Kauffman bracket skein algebra and module of the 3-twist knot complement.
result Determine the action of Kauffman bracket skein algebra on module of 3-twist knot complement.
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…