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 \( n \).
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…
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…
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, …
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.
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.
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.
For Legendrian links in the 1-jet space of S1 we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kau…
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.
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.
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…
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 …
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.
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
New formula proves skein modules are finite for 3-manifolds.
problem Proving skein modules are finite for closed 3-manifolds.
method Using Heegaard splittings and algebraic computation.
result Skein modules are finite-dimensional, resolving a conjecture.
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
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 Z[A±1]. result The skein module of (S1imesS2) # (S1imesS2) does not split into free and torsion submodules. 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…
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.
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.
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…
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…
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 gl2-skein modules for lens spaces.
problem Computing gl2-skein modules for lens spaces. method Action of gl2-skein algebra on solid torus's gl2-skein module. result Lens spaces' gl2-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.
Study Type C skein modules using Sp(2n) webs and construct transparent elements.
problem Understanding Type C skein modules and constructing transparent elements. method Diagrammatic approach using multivariable Chebyshev polynomials and explicit braiding formulas.
result Construction of transparent elements in the skein module at roots of unity.
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
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 q and decomposing twisted Hochschild homology of G-skein algebras. result Dimensions of skein modules for G=SL2 and G=GL1 are calculated. 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 shows skein module dimensions for surface times circle.
problem Understanding skein modules of surface times circle.
method Used Kauffman bracket skein module over rational functions.
result Dimension at least 2^{2g+1}+2g-1 for Sigma x S^1.
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.
New definition of skein lasagna module for specific 4-manifolds.
problem Defining a new homology theory for specific 4-manifolds.
method Elementary definition using diagrams in the 4-manifold.
result Explicit calculations for disk bundles over S^2.
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
Alternative basis for Kauffman bracket skein module of solid torus found using braids.
problem Computing Kauffman bracket skein module of lens spaces.
method Using Temperley--Lieb algebra of type B and braids.
result Alternative basis BmST for ${
m KBSM}\left({
m ST}
ight)$. Study on skein modules and character varieties of Seifert manifolds.
problem Characterizing and analyzing skein modules and character varieties of Seifert manifolds.
method Analyzing character varieties and skein modules of Seifert manifolds, computing dimensions and showing their equivalence.
result Finitely generated skein modules and reduced character varieties for certain Seifert manifolds.
Study of skein modules in 3-manifolds, showing non-injectivity results.
problem Understanding skein modules in 3-manifolds and their behavior under gluing.
method Extended Kauffman bracket skein modules to 3-manifolds with marking, introduced new maps and studied their properties.
result Non-injectivity of certain maps in stated skein modules, especially when quantum parameter is a root of 1.
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. 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.
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…