This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of t…
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48 results for “equivariant skein relations”
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.
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.
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.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
Constructs a TQFT for 3-manifolds and extends it to 4-dimensional 2-handlebodies.
problem Building a TQFT for 3-manifolds and extending it to 4D.
method Using a Frobenius algebra and skein relations, constructing a TQFT for 3-manifolds and extending it to 4D using an inductive state-sum construction.
result Extends a TQFT to 4-dimensional 2-handlebodies.
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
Let be a trivial knot in the three-sphere. For every finite cyclic group of odd order, we construct a -equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to isotopy in . Another interpretation is given using the categorification of the …
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed link homology.
result Non-vanishing and decomposition results for deformed skein lasagna modules.
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.
On skein invariantsmath.GT
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.
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.
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.
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant …
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY …
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.
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
Paper introduces a new skein relation for multivariable polynomials of virtual links.
problem Understanding properties of virtual links through polynomial invariants.
method Developed a virtual skein relation for multivariable polynomials of virtual links.
result New skein relation for multivariable polynomials of virtual links.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
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.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
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…
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.
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…
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-a…
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.
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's quantum invariants. Our method consists in the study of Mark…
In this note we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.
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 Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…
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 …
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use…
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.
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
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.
Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariant. This invariant can also be …
New categories from TQFTs interpret skein relations.
problem Interpreting skein relations in TQFTs.
method Constructing half-braided algebras and their bimodules.
result Stated skein relations correspond to a TQFT.
For a ring , we denote by the free -module spanned by the isotopy classes of singular links in . Given two invertible elements , the HOMFLY-PT skein module of singular links in (relative to the triple ) is the quotient of by local rela…
The study provides a basis for a 3-manifold's skein module, answering its dimension.
problem Determining the dimension of the Kauffman skein module of a surface times a circle.
method Explicitly spanning family for the skein modules provided for any closed oriented surface .
result The dimension of is .
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.
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.
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 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.
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.
Kauffman bracket skein algebra structure on surfaces defined.
problem Understanding the structure of Kauffman bracket skein algebra for surfaces.
method Associated generating sets and relations of degree at most 6 for embedded graphs.
result Ideal of defining relations generated by specific subsurfaces.
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.
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].