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.
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
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.
Paper proves zero stability for one-row colored sl₃-Jones polynomials.
problem Stability of coefficients in colored Jones polynomials.
method Linear skein theory based on Kuperberg's sl₃-webs.
result Zero stability for B-adequate links with anti-parallel twist regions.
Study of Chern-Simons theory and link invariants using gauge fields and skein relations.
problem Understanding link invariants and their relations to Chern-Simons theory.
method Differential-geometric methods to extend functional integrals and skein relations for Wilson loops.
result Skein relations for link invariants are shown to be related to the Kauffman bracket polynomial and Jones polynomial.
We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one should start from knots in 3-manifolds, surface…
Homflypt skein theory and string topology linked via 2-groupoids.
problem Understanding relations in Homflypt skein theory.
method Defined a 2-groupoid from the fundamental 2-groupoid of singular links, relating it to string topology.
result Relations in Homflypt skein theory are induced from a 2-groupoid.
The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each …
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.
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. 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.
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.
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. The paper explores basic properties of knot skein invariants.
problem Understanding skein invariants of knots.
method Discussion of basic properties and known examples of skein invariants.
result Discussion of basic properties and known examples of skein invariants.
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
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.
Extends 4D cornered skein theory to surfaces, proving gluing formulas.
problem Formulating gluing formulas for 4-manifolds with corners and boundaries.
method Develops a categorical framework and introduces bicategories for closed surfaces.
result Proves gluing formulas for categories associated with 3-manifolds with boundary.
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
New skein theory for Links-Gould polynomial simplifies link evaluations.
problem Computing Links-Gould polynomial for oriented links.
method Developed a cubic braid-type skein theory.
result Skein theory can evaluate any oriented link.
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.
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.
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.
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 Aq polynomial. Researchers prove positivity of skein algebra structure constants for specific surfaces.
problem Positivity of structure constants in skein algebras of specific surfaces.
method Mirror symmetry construction based on higher genus Gromov-Witten theory applied to a complex cubic surface.
result Proved positivity of structure constants for skein algebras of the 4-punctured sphere and 1-punctured torus.
The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.
problem Classifying virtual knot polynomials and trivalent graph invariants with specific conditions.
method Skein-theoretic techniques applied to classify invariants with smallness conditions.
result Classification of all non-trivial invariants of trivalent graphs and skein theories of virtual tangles.
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.
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…
The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…
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.
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.
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.
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 skein modules via gauge theory, finding non-TQFT dimensions.
problem Understanding skein modules of 3-manifolds using gauge theory.
method Embed skein modules into 4d N=4 super-Yang-Mills theories, using N=1 supersymmetry. result Find non-standard dimensions of skein modules, differing from TQFT predictions.
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 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out using Mathematica, Maple, Mathcad, etc.
The study extends knot theory to knotoids using two approaches.
problem Extending Vassiliev invariants to knotoids.
method Two approaches: 1) Closures to knots, 2) Directly on knotoids.
result Non-trivial type-1 invariants for spherical knotoids.
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
Develops skein theory for 3-manifolds with defects, extending quantum character stacks.
problem Quantum character stacks and their applications in 3-manifolds with surface defects.
method Parabolic induction/restriction for quantum groups, quantum decorated character stacks, ideal triangulations, gluing equations.
result Knot invariants related to quantum A-polynomial, concrete computation method. In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.
Defines map from skein module to Habiro ring for quantum modularity.
problem Quantum modularity conjecture and its arithmetic aspects.
method Defining a map from Kauffman bracket skein module to Habiro ring using Witten's conjecture.
result Image is an effectively computable module of finite rank.
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.
New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
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.
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …
The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.
problem Establishing Tait conjectures for adequate links in thickened surfaces.
method Applying Kauffman bracket skein algebras to develop a theory of skein adequate links and proving Tait conjectures.
result The crossing number is additive under connected sum for adequate links in thickened surfaces.
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.
Skein theory aids in computing YB homology and cohomology groups.
problem Computing Yang-Baxter homology and cohomology groups.
method Skein theoretic approach to decompose the YB operator and its differentials.
result Computed second and third YB homology groups, confirming a conjecture.
Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable …