Proves method to compute HOMFLYPT skein module of lens spaces L(p,1) from solid torus.
problem Computing HOMFLYPT skein module of lens spaces L(p,1) from solid torus. method Solving infinite system of equations involving Lambropoulou invariant and braid band moves.
result Derives HOMFLYPT skein module of lens spaces L(p,1) from solid torus. 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.
This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Marko…
Researchers compute HOMFLYPT skein module of lens spaces using braids.
problem Computing the HOMFLYPT skein module of lens spaces L(p,1). method Using braids to solve an infinite system of equations.
result Established the relation between S(L(p,1)) and S(ST). 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.
Researchers compute the skein module of a solid torus using braids.
problem Computing the HOMFLYPT skein module of S1imesS2. method Braid-theoretic techniques to solve an infinite system of equations.
result The free part of the skein module is generated by the empty link, with all other elements being torsion.
If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the simila…
This paper computes the Homflypt skein module of lens spaces using braids.
problem Computing the Homflypt skein module of lens spaces L(p,1). method Using Lambropoulou invariant and braid band moves on a basis of the solid torus.
result Established the connection and computed the Homflypt skein module of lens spaces.
An invariant for knots with colored bonds respects HOMFLYPT relation.
problem Modeling and distinguishing knots with colored bonds.
method Introducing a HOMFLYPT skein module for colored bonded knots.
result The non-rigid version of the module provides information about knottedness of bonds.
Survey on link diagrams in Seifert manifolds and skein modules.
problem Understanding link invariants in Seifert manifolds.
method Use of arrow diagrams and Reidemeister moves.
result New bases of skein modules for specific Seifert manifolds.
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.
Survey of knot invariants in lens spaces.
problem Comparing knot invariants in lens spaces.
method Summarized various invariants and compared their distinguishing power.
result Invariants generalize classical knot polynomials and can distinguish difficult cases.
Researchers develop a new basis to simplify solving infinite systems for HOMFLYPT skein module of lens spaces.
problem Computing the HOMFLYPT skein module of lens spaces L(p,1) via braids. method Using a new basis Λ of the HOMFLYPT skein module of the solid torus, relating infinite systems of equations obtained by performing braid band moves on elements in Λ+ and Λ− via a map I. result Reduced complexity in solving the infinite system of equations for L(p,1) using braids. Let k be a subring of the field of rational functions in x,v,s which contains x±1,v±1,s±1. If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the…
Mathematical study connects curve counting to quantum topology.
problem Counting holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds.
method Organized 1-parameter families of holomorphic curves and matched them to HOMFLYPT skein relations.
result Holomorphic curve counts match HOMFLYPT polynomial coefficients of links in the 3-sphere.
Following the recent work by Chan, and by Morton and Hadji on the Homflypt polynomials of some generalized Hopf links, we investigate the Kauffman polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S^1\times D^2, we establish a similar skein map on the Kauffman skein module …
Let k be an integral domain containing the invertible elements α, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein mod…
Proves strong integrality for colored HOMFLYPT invariants.
problem Proving strong integrality for colored HOMFLYPT invariants.
method Using HOMFLY skein theory.
result Strong integrality theorem for reduced colored HOMFLYPT invariants.
Paper introduces equations to distinguish knots without using knot invariants.
problem Distinguishing knots using only topological and combinatorial methods.
method Linear systems of equations derived from HOMFLYPT and Kauffman polynomials.
result First examples of distinguishing knots without knot invariants.
The Homflypt and Kauffman skein modules of the projective space are computed. Both are free and generated by some infinite set of links. This set may be chosen to be L_n, where L_n is an arbitrary link consisting of n projective lines for n>0, and L_0 is an affine unknot.
In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus C. We show that the full colored HOMFLYPT invariant has a nice structure when q→1. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to …
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 constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
New knot invariants created using multiple skein relations.
problem Creating more powerful knot invariants.
method Introducing new ways to smooth crossings and using a system of skein equations.
result A simplified version of the modified invariant is easier to compute and generalizes the two-variable Kauffman polynomial.
Paper shows links with same Homflypt but different Conway type invariant.
problem Comparing Homflypt and Conway type invariants for links.
method Applied two skein relations to crossings to define generalized Conway type invariant.
result Found links with same Homflypt but different Conway type invariant.
The paper studies Betti numbers of HOMFLYPT homology modules for links.
problem Understanding the structure of HOMFLYPT homology modules and their Betti numbers.
method Analyzing the module structure and Betti numbers of the middle HOMFLYPT homology for links.
result The Betti numbers of the middle HOMFLYPT homology modules are finite and can be used to recover the Poincaré polynomial.
New skein invariants defined for links using a novel procedure.
problem Defining new skein invariants for links.
method Applying skein relation to distinct component crossings, then evaluating resulting knots with given invariants.
result Generalizations of known skein invariants for classical links.
Algorithm calculates polynomial coefficients of link invariants.
problem Computing first coefficients of link invariants.
method Dynamic programming algorithm for Homflypt and Kauffman polynomials.
result Polynomial time complexity for first coefficients.
In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), which was predicted by Jozef Przytycki, using topological interpretation. The basis Λ is different from the basis Λ′, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} w…
New invariants of links are constructed using the skein invariant polynomial of colored links defined by the author in [1]. These invariants are stronger than the homflypt polynomial.
New identities lift q-dilogarithm to a more complex algebra.
problem Extending q-dilogarithm identities to a more complex algebra.
method Showed lift to HOMFLYPT-skein algebra of a genus n handlebody.
result New identities associated to unidirectional A_n-quiver.
In this paper we announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type A. Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the fa…
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.
Computes colored HOMFLYPT invariants using holomorphic curves.
problem Counting holomorphic curves in Calabi-Yau 3-folds.
method Computes contributions of multiple covers of holomorphic annuli.
result Agrees with topological string theory predictions and proves Ooguri-Vafa formula.
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.
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.
New generators identified for 3-torus skein module.
problem Identifying generators for Kauffman bracket skein module of 3-torus.
method Showed linear independence of 9 identified generators.
result Linearly independent set of generators for 3-torus 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 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…
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.