New grading on algebras of curves by winding number.
problem Understanding the structure of algebras of curves.
method Constructing a new grading on the Goldman Lie algebra and related algebras by winding number.
result Induces a new grading on the HOMFLY-PT skein algebra and related algebras.
Formula for Dehn twists on HOMFLY-PT skein modules with applications.
problem Understanding the action of Dehn twists on HOMFLY-PT skein modules.
method Introduced a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface.
result Constructed an invariant \( z(M) \) for integral homology 3-spheres, finite type invariant of order \( n \).
New method computes automorphisms of surface groups using skein algebras.
problem Computing automorphisms of surface groups.
method Using skein algebras and Goldman Lie algebra.
result Refined formula for automorphisms of homology cylinders.
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
The paper introduces new structures for colored HOMFLY-PT invariants using skein theory.
problem Proving strong integrality and deriving symmetric properties for HOMFLY-PT invariants.
method Purely using HOMFLY-PT skein theory and applying to LMOV conjecture.
result Strong integrality and symmetric properties for colored HOMFLY-PT invariants.
For a ring R, we denote by R[L] the free R-module spanned by the isotopy classes of singular links in S3. Given two invertible elements x,t∈R, the HOMFLY-PT skein module of singular links in S3 (relative to the triple (R,t,x)) is the quotient of R[L] by local rela…
In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in S1×S2 which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT ske…
We show that for any Legendrian link L in the 1-jet space of S1 the 2-graded ruling polynomial, RL2(z), is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover RL2(z) as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, …
A new polynomial invariant for strongly involutive links.
problem Characterizing strongly involutive links using polynomial invariants.
method Introducing a two-variable polynomial invariant \(P^e\) with equivariant skein relations.
result Specialisation of \(P^e\) recovers the graded Euler characteristic of a spectral sequence.
The paper improves bounds on the complexity of computing link polynomials.
problem Computing link polynomials by the skein relation is complex.
method Proved new upper and lower bounds on skein tree depth.
result New bounds on skein tree depth are stronger than previous ones.
The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.
problem Improving bounds on skein tree depth and delta-crossing numbers for knots and links.
method Theoretical and computational analysis of skein trees and knot invariants.
result New upper and lower bounds on skein tree depth and delta-crossing numbers are derived.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
Extends A-type coefficient polynomials to B-type setting, introducing new invariants.
problem Tackles the B-type skein relation and introduces new coefficient polynomials.
method Introduces coefficient polynomials associated with the B-type skein relation and proves their invariance under Reidemeister moves.
result Shows that the generating series of these coefficient polynomials recovers the Kauffman polynomial.
We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries…
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
Decomposes SL3 skein algebras for surfaces.
problem Decomposing SL3 skein algebras for surfaces. method Splitting surfaces into triangles and analyzing the resulting algebras.
result Explicit basis and injective splitting morphisms for SL3 stated skein algebras. Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of (n,n+1) torus knots. method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
Survey on stated skein algebras and their representations.
problem None explicitly stated in the abstract.
method None explicitly stated in the abstract.
result None explicitly stated in the abstract.
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.
Paper presents skein algebras for spheres with punctures.
problem Quantization of decorated Teichmüller space.
method Presentations of Roger-Yang generalized skein algebras for punctured spheres.
result New interpretation of homogeneous coordinate ring of Grassmannian of planes.
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. New invariant derived from skein algebra representations at roots of unity.
problem Understanding skein algebras of open surfaces and their invariants.
method Extended skein algebras, constructed isomorphisms, embedding and isomorphism constructions.
result Invariant associated to each skein algebra representation class.
New inductive basis proves Markov trace existence and transverse traces determination.
problem Existence and characterization of traces on Birman-Murakami-Wenzl algebras.
method Inductive basis construction and proof of trace existence and determination.
result All transverse traces are determined by specific polynomials and link invariants.
This paper calculates the skein algebra of the Borromean rings complement.
problem Calculating the skein algebra of the Borromean rings complement.
method Using the skein algebra definition and character variety, the polynomial ring quotient is determined.
result An explicit formula for the skein algebra of the Borromean rings complement is provided.
Generators of skein algebra of surfaces are algebraic curves.
problem Describe generators of the skein algebra of surfaces.
method Prove algebraic generators are simple closed curves associated to mapping class group generators.
result Conjecture a presentation for the skein algebra of surfaces.
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.
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.
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.
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.
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].
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.
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.
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.
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.
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 connects two skein algebras and characterizes their representations.
problem Characterizing representations of Roger-Yang skein algebras.
method Calculating Roger-Yang skein algebra of an annulus, establishing a homomorphism to Kauffman bracket skein algebra of a torus, and using these to characterize representations.
result Characterization of irreducible, finite-dimensional representations of Roger-Yang skein algebra of an annulus with two interior punctures.
Quantum torus methods enhance understanding of skein algebras and modules.
problem Investigating Kauffman bracket skein algebras and modules using quantum torus methods.
method Two quantum torus methods: embedding into quantum Teichmüller space and filtering to a monomial subalgebra.
result Generalized Chebyshev homomorphism and refined unicity theorem for skein modules.
Method computes centers of Poisson and skein algebras for loops on surfaces.
problem Computing centers of Poisson and skein algebras associated to loops on surfaces.
method Systematic method using Goldman and Wolpert's Poisson algebras and Turaev's skein algebras.
result Computed centers of various Poisson and skein algebras for finite type hyperbolic surfaces.
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.
The paper studies algebraic and geometric properties of stated skein algebras of surfaces.
problem Understanding the algebraic and geometric properties of stated skein algebras of surfaces.
method Analyzes the skein algebra of surfaces, proving isomorphisms and lifting properties, and interpreting topologically.
result The skein algebra of a surface with n boundary components is an algebra-comodule over Oq2(SL(2))⊗n. The paper proves unicity for Kauffman bracket skein algebras.
problem Unicity conjecture for Kauffman bracket skein algebras of surfaces.
method General unicity theorem applied to Kauffman bracket skein algebras, center characterization, and finitely generated module over center.
result Irreducible representations of Kauffman bracket skein algebras are classified by their central characters.
Monoidal categorifies genus zero skein algebra using K-theory.
problem Relating skein algebra to K-theory.
method Using quantized K-theoretic Coulomb branch and Grothendieck ring of equivariant coherent sheaves.
result Monoidal categorification of the 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.
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. Proves a pentagon relation in skein theory.
problem Developed a skein-theoretic version of cluster theory and conjectured a pentagon relation.
method Topological proof using skein algebra and elliptic Hall algebra.
result Proves the pentagon relation for the skein dilogarithm.
Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
problem Injectivity of Roger-Yang's homomorphism and zero divisors in skein algebra.
method Hyperbolic geometric considerations, ideal triangulation, normal arcs.
result Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.
problem Cluster and skein algebras on surfaces with punctures.
method Geometric and algebraic methods, including decorated Teichmüller spaces and skein algebras.
result Cluster and skein algebras coincide for surfaces with at least 2 punctures.