New sl(2) action defined on a mathematical module.
problem No specific problem stated; focuses on mathematical construction.
method Construction of sl(2)-action on equivariant skein lasagna module.
result Infinitesimal sl(2)-symmetries constructed.
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48 results for “SL_n skein modules”
Quantum Frobenius map for skein modules constructed and described.
problem Constructing a quantum Frobenius map for skein modules.
method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group
result Described the quantum Frobenius map for skein modules.
Researchers calculate dimensions of skein modules for 2-torus mapping tori.
problem Determining dimensions of Kauffman bracket skein modules for specific cases.
method Using generic and decomposing twisted Hochschild homology of -skein algebras.
result Dimensions of skein modules for and are calculated.
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 and provides geometric interpretations of quantum group structures.
Novel method computes Kauffman bracket skein modules of small 3-manifolds.
problem Computing Kauffman bracket skein modules of small 3-manifolds.
method Developed a novel method to compute these modules.
result Dimension of skein modules of 3-manifolds related to character varieties and Abouzaid-Manolescu invariants.
We show that we can release the rigidity of the skew Howe duality process for 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 case, corresponding to looking at tan…
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as polynomial.
Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…
Frobenius homomorphisms for SL_n skein modules generalize knot theory results.
problem Quantum group representations and skein theory for SL_n character varieties.
method Representation theory of quantum groups and skein theory.
result Frobenius homomorphisms for stated SL_n skein modules are defined and their properties are explored.
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.
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 and clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…
Establishes a connection between skein modules and algebraic sets.
problem Understanding the structure of stated -skein modules.
method Uses algebraic homomorphisms and isomorphisms to relate skein modules to algebraic structures.
result Proves the isomorphism between stated skein modules and universal representation algebras.
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
Quantum Character Theorymath.RT
Develops quantum character theory for complex reductive groups.
problem Quantum analogue of conjugation equivariant -modules.
method Schur-Weyl functor and double affine Hecke algebra.
result Computes endomorphism algebras of quantum Hotta-Kashiwara modules.
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.
Study torsion in Kauffman bracket skein modules of 3-manifolds.
problem Identify compact oriented 3-manifolds with torsion in their Kauffman bracket skein modules.
method New criteria based on -character varieties and effective criteria for manifolds with incompressible tori.
result Many counterexamples to Kirby's list and new criteria for torsion presence.
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.
The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Spider category comparison proves equivalence to Sikora's quotient category.
problem Comparing skein theories of .
method Proved equivalence between spider category and Sikora's quotient category.
result Spider category is equivalent to Sikora's quotient category.
The Kauffman bracket skein modules, S(M,A), have been calculated for A=+1,-1, for all 3-manifolds M by relating them to the SL(2,C)-character varieties. We extend this description to the case when A is a 4-th root of 1 and M is either a surface x [0,1] or a rational homology sphere (or its submanifold).
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…
Skeins on torimath.QA
Study of skein invariants on tori for various groups and quantum parameters.
problem Analysis of -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.
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.
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.
Decomposes skein algebras for surfaces.
problem Decomposing skein algebras for surfaces.
method Splitting surfaces into triangles and analyzing the resulting algebras.
result Explicit basis and injective splitting morphisms for stated skein algebras.
The paper defines a new algebra structure on skein modules of 3-manifolds and shows it's isomorphic to a known algebra.
problem Defining and studying algebraic structures on skein modules of 3-manifolds.
method Introducing and analyzing a new algebra structure based on links in -manifolds.
result The algebra is naturally isomorphic to a known algebra when the parameter is .
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
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…
Extremal weight projectorsmath.RT
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 study proves representations for certain 3-manifolds using gauge theory.
problem Proving irreducible SL(2,C) representations for specific 3-manifolds.
method Instanton gauge theory applied to 3-manifolds with specific torsion properties.
result Irreducible SL(2,C) representations exist for certain 3-manifolds.
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
Admissible Skein Modulesmath.GT
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.
Higher Skein Modulesmath.GT
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
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.
Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…
Researchers compute the Kauffman bracket skein module of a specific 3-manifold.
problem Understanding the structure of Kauffman bracket skein modules for non-prime manifolds.
method Analyzing handle sliding relations to compute the module over .
result The skein module of does not split into free and torsion submodules.
Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
problem Understanding the genus-two handlebody skein module.
method Using q-difference operators for the genus-two skein algebra.
result Correspondence between reduced Askey-Wilson polynomials and genus-two handlebody skein module.
Study of cubic skein modules in 3-sphere and arbitrary 3-manifolds.
problem Lack of systematic study of higher degree skein modules.
method Investigation of cubic skein module structure and properties in 3-sphere and arbitrary 3-manifolds.
result Establishment of a foundational framework for higher skein modules.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…
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.
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
Skein lasagna module calculates 4-manifold invariants using handle decompositions.
problem Calculating invariants of 4-manifolds and links.
method Express skein lasagna module in terms of handle decompositions.
result Skein lasagna module can be locally infinite dimensional.
Researchers compute -skein modules for lens spaces.
problem Computing -skein modules for lens spaces.
method Action of -skein algebra on solid torus's -skein module.
result Lens spaces' -skein modules span by specific elements.
Study on skein module dimensions at irreducible representations.
problem Dimension of skein module at irreducible representations.
method Localization of skein module at maximal ideal corresponding to irreducible representation.
result Localization forms a one-dimensional free module over the unreduced coordinate ring.
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
Study Kauffman bracket skein modules of Seifert fibered spaces.
problem Understanding the structure of Kauffman bracket skein modules.
method Investigate spanning sets and module structure.
result Kauffman bracket skein modules are finitely generated.